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ELEMENTS 



OF 



MECHANICS 



TREATED BY MEANS OF 



THE DIFFERENTIAL AND INTEGRAL 
CALCULUS. 



BY 



WILLIAM G. PECK, Ph.D., LL.D., 

PROFESSOR OF MATHEMATICS, ASTRONOMY, AND MECHANICS, COLUMBIA 
COLLEGE, 







A. S. BARNES & COMPANY 

NEW YORK AND CHICAGO 



Copyright, 1859, by "Wm. G. Peck.. 
Copyright renewed, 1887, by Wm. G. Peck. 



PREFACE 



The following work was undertaken to supply a want felt 
by the author, when engaged in teaching Natural Philosophy 
to College classes. In selecting a text-book on the subject 
of Mechaotcs, there was no want of material from which to 
choose ; but to find one of the exact grade for College 
instruction, was a matter of much difficulty. The higher 
treatises were found too difficult to be read with profit, 
except by a few in each class, in addition to which they 
were too extensive to be studied, even by the few, in the 
limited time allotted to this branch of education. The 
simpler treatises were found too elementary for advanced 
classes, and on account of their non-mathematical character, 
not adapted to prepare the student for subsequent investi- 
gations in Science. 

The present volume was intended to occupy the middle 
ground between these two classes of works, and to form a 
connecting link between the Elementary and the Higher 
Treatises. It was designed to embrace all of the important 
propositions of Elementary Mechanics, arranged in logical 
order, and each rigidly demonstrated. If these designs 



IV PREFACE. 

have been accomplished, this volume can be read with 
facility and advantage, not only by College classes, bat by 
the higher classes in Academies and High Schools ; it will 
be found to contain a sufficient amount of information for 
those who want either the leisure or the desire to make 
the mathematical sciences a specialty ; and finally, it will 
serve as a suitable introduction to those higher treatises on 
Mechanical Philosophy, which all must study who would 
appreciate and keep pace with the wonderful discoveries 
that are daily being made in Science. 

Columbia College, February 22, 185$ 



PREFACE TO THE SECOND EDITION. 



In accordance with the expressed wish of many teach- 
ers in institutions where the Differential and Integral 
Calculus are either not taught at all, or else are not 
obligatory studies, an Appendix has been added to the 
body of the work, in which all of the principles there 
demonstrated by means of the Calculus are deduced by 
the aid of Elementary Mathematics only. 

It has not seemed desirable to omit the Calculus 
altogether, especially as by the present arrangement the 
work is equally adapted to the use of those who teach 
by the aid of the Calculus, and of those who only em- 
ploy the Elementary Mathematics. 

From the flattering reception of this work by the 
P iblic, it is believed that a continuation of the Course 
of Natural Philosophy, of which this is the opening vol- 
ume, would be acceptable. To carry out this design, 



VI » PREFACE. 

two other volumes are in preparation on the same gen. 
eral plan as the present, one of which will be devoted 
to the subjects of Acoustics and Optics, and the other 
to Heat and the Steam-Engine, Electricity \ and Mag- 
netism. 

February 22, I860* 



CONTENTS 



CHAPTER I. 

PAcm 

Definitions — Rest and Motion 13 

Forces 14 

Gravity 15 

Weight— Mass 16 

Momentum — Properties of Bodies 17 

Definition of Mechanics — Measure of Forces 2: 

Representation of Forces 23 



CHAPTER II. 

Composition of Forces whose Directions coincide 25 

Parallelogram of Forces 26 

Parallelopipedon of Forces. ... 27 

Geometrical Composition and Resolution of Forces 28 

Components in the Direction of two Axes 30 

Components in the Direction of three Axes 32 

Projection of Forces 34 

Composition of a Group of Forces in a Plane 35 

Composition of a Group of Forces in Space 36 

Expression for the Resultant of two Forces 37 

Principle of Moments 40 

Principle of Virtual Moments '43 

vii 



Vlll CONTENTS. 

PAGR 

Resultant of Parallel Forces 45 

Composition and Resolution — Parallel Porces 48 

Lever arm of the Resultant 51 

Centre of Parallel Forces 52 

Resultant of a Group in a Plane 53 

Tendency to Rotation — Equilibrium in a Plane 58 

Equilibrium of Forces in Space 59 

Equilibrium of a Revolving Body 6C 



CHAPTER III. 

Weight — Centre of Gravity 62 

Centre of Gravity of Straight Line 64 

Of Symmetrical Lines and Areas 64 

Of a Triangle 65 

Of a Parallelogram — Of a Trapezoid 66 

Of a Polygon 67 

Of a Pyramid 68 

Of Prisms, Cylinders, and Polyhedrons , 70 

Centre of Gravity Experimentally 71 

Centre of Gravity by means of the Calculus 72 

Centre of Gravity of an Arc of a Circle 73 

Of a Parabolic Area 74 

Of a Semi-Ellipsoid. 75 

Pressure and Stability 80 

Problems in Construction 85 



CHAPTER IT. 

Definition of a Machine ... 94 

Elementary Machines — Cord 96 

The Lever 98 

The Compound Lever 101 

The Elbow-joint Press 102 

The Balance 103 






CONTENTS. IX 

PAGE, 

The Steelyard 105 

The Bent Lever Balance— Compound Balances 106 

The Inclined Plane. HO 

The Pulley 112 

Single Pulley 113 

Combinations of Pulleys 115 

The Wheel and Axle 117 

Combinations of Wheels and Axles 118 

The Windlass '. 119 

The Capstan— The Differential Windlass 120 

Wheel-work 121 

The Screw 123 

The Differential Screw : 125 

Endless Screw 126 

The Wedge 127 

General Remarks on Machines 129 

Friction 130 

Limiting Angle of Resistance 133 

Rolling Friction — Adhesion 135 

Stiffness of Cords 136 

Atmospheric Resistance —Friction on Inclined Planes 137 

Line of least Fraction 140 

Friction on Axle 141 



CHAPTER Y. 

Uniform Motion 143 

Varied Motion 144 

Uniformly Varied Motion ; 146 

Application to Falling Bodies 143 

Bodies Projected Upwards 15C 

Restrained Vertical Motion 153 

Atwood's Machines 156 

Motion on Inclined Planes 158 

Motion down a Succession of Inclined* Planes 161 

Periodic Motion ] 63 



X CONTENTS. 

TACHL 

Angular Telocity.. . . .'....„ . . . ; 16c 

The Simple Pendulum 166 

The Compound Pendulum 169 

Practical Applications of the Pendulum 1*75 

Graham's and Harrison's Pendulums 1*76 

Basis of a System of Weights and Measures 177 

Centre of Percussion 179 

Moment of Inertia 180 

Application of Calculus to Moment of Inertia 182 

Centre of Gyration 186 



CHAPTER VI. 

Motion of Projectiles 188 

Centripetal and Centrifugal Forces 197 

Measure of Centrifugal Force 197 

Centrifugal Force of Extended Masses 203 

Principal Axes 206 

Experimental Illustrations 207 

Elevation of the outer rail of a Curved track 209 

The Conical Pendulum ' 210 

The Governor 212 

Work 215 

Work, when the Power acts obliquely 217 

Work, when the Body moves on a Curve 219 

Rotation — Quantity of Work „ 22S 

Accumulation of Work 225 

Living Force of Revolving Bodies 227 

Fly Wheels 228 

Composition of Rotations , 230 

Application to Gyroscope 232 

CHAPTER VII. 

Classification of Fluids 236 

Principle of Equal Pressures 236 



CONTENTS. XI 

PAGE. 

Pressure due to Weight 238 

Centre of Pressure on a Plane Surface 243 

Buoyant Effect of Fluids 249 

Floating Bodies 249 

Specific Gravity 251 

Hydrostatic Balance 253 

Specific Gravity of an Insoluble Body 253 

Specific Gravity of Liquids 254 

Specific Gravity of Soluble Bodies 255 

Specific Gravity of Air and Gases. 256 

Hydrometers — Nicholson's Hydrometer 257 

Scale Areometer 258 

Volumeter 259 

Densimeter 260 

Centesimal Alcoholometer of Gay Lussac 261 

Thermometer 263 

Velocity of a Liquid through an Orifice 265 

Spouting of Liquids on Horizontal Planes 268 

Modifications due to Pressures 269 

Coefiicients of Efflux and Velocity 210 

Efflux through short Tubes 272 

Motion of Water in open Channels 274 

Motion of Water in Pipes 277 

General Remarks 27 9 

Capillary Phenomena 280 

Elevation and Depression between Plates 281 

Attraction and Repulsion of Floating Bodies 282 

Applications of the principle of Capillarity 283 

<£ndosmose and Ezosmose 284 



CHAPTER VIII. 

Gases and Vapors 285 

Atmospheric Air 285 

Atmospheric Pressure 286 

Marictte's Law 287 



Xll CONTENTS. 

PAGE 

Gay Lussac's Law 290 

Manometers — The open Manometer 291 

The closed Manometer 292 

The Siphon Guage 294 

The Barometer — Siphon Barometer 295 

The Cistern Barometer 296 

Uses of the Barometer 297 

Difference of Level 298 

Work of Expanding Gas or Vapor 304 

Efflux of a Gas or Vapor 306 

Steam 308 

Work of Steam 310 

Experimental Formulas 311 



CHAPTER IX. 

Pumps — Sucking and Lifting Pumps 313 

Sucking and Forcing Pump. 318 

Fire Engine 321 

The Rotary Pump 322 

Hydrostatic Press 324 

The Siphon 326 

Wurtemburg and Intermitting Siphon 328 

Intermitting Springs 328 

Siphon of Constant Flow — Hydraulic Ram 329 

Archimedes' Screw 331 

The Chain Pump— The Air Pump 332 

Artificial Fountains— Hero's Ball 336 

Hero's Fountain 837 

Wine-Taster and Dropping Bottle 338 

The Atmospheric Inkstand S3? 



MECHANICS. 



CHAPTER I. 



DEFINITIONS AND INTRODUCTORY REMARKS. 

Definition of Natural Philosophy. 

1. Natural Philosophy is that branch of Science which 
treats of the laws of the material universe. 

These laws are called laws of nature ; and it is assumed 
that they are constant, that is, that like causes always pro- 
duce like effects. This principle, which is the basis of all 
Science, is an inductive truth founded upon universal experi- 
ence. 

Definition of a Body. 

2. A Body is a collection of material particles. When 

the dimensions of a body are exceedingly small, it is called 

a material point. 

Rest and Motion. 

3. A body is at rest when it retains the same absolute 
position in space ; it is in motion when it continually 
changes its position. 

A body is at rest with respect to surrounding objects, 
when it retams the same relative position with respect to 
them ; it is in motion with respect to them, when it con- 
tinually changes this relative position. These states are 
called relative rest and relative motion, to distinguish them 
from absolute rest and absolute motion. It is highly prob- 
able that no object in the universe is in a state of absolute 
rest. 



M- MECHANICS. 

Trajectory. 

4. The path traced out, or described by a moving poiit. 
is called its trajectory. When this trajectory is a straight 
line, the motion is rectilinear ; when it is a curved line, the 
motion is curvilinear. 

Translation and Rotation. 

5. When all of the points of a body move in parallel 

straight lines, the motion is called motion of translation ; 

when the points of a body describe arcs of circles about a 

straight line, the motion is called motion of rotation. 

Other varieties of motion result from a combination of 

these two. 

Uniform and Varied Motion. 

6. The velocity of a moving point, is its rate of motion. 

When the point moves over equal spaces in any arbitrary 

equal portions of time, the motion is uniform, and the 

velocity is constant y when it moves over unequal spaces in 

equal portions of time, the motion is varied, and the velocity 

is variable. If the velocity continually increases, the motion 

is accelerated ; if it continually decreases, the motion is 

retarded. 

Forces. 

7. A Foece is anything which tends to change the state 
of a body with respect to rest or motion. 

If a body is at rest, anything which tends to put it in 
motion is a force ; if it is in motion, anything which tends 
to make it move faster, or slower, is a force. The power 
with which a force acts, is called, its intensity. 

Forces are of two kinds : extraneous, those which act upon 
a body from without ; molecular, those which are exerted 
between adjacent particles of bodies. 

An extraneous force may act for an instant and then cease, 
in which case it is called an impulse, or an impulsive force / 
or it may act continuously, in which case it is called an 
incessant force. An incessant force may be regarded as 
made up of a succession of impulses acting at equal but 
exceedingly small intervals of time. When these successive 



DEFINITIONS AND INTRODUCTORY REMARKS. 15 

impulses are equal, the force is constant ; when they are 
unequal, the force is variable. The force of gravity at any 
given place, is an example of a constant force ; the effort of 
expanding steam, is an example of a variable force. 

Molecular forces are of two kinds; attractive, those which 
tend to draw particles together ; repellent, those which tend 
to separate them. These forces also exert an arranging 
power by virtue of which the particles of bodies are grouped 
into definite shapes. The phenomena of crystalization pre- 
sent examples of this action. Molecular forces of both kinds 
are continually exerted between the particles of all bodies, 
and upon their variation, in intensity and direction, depend 
the conditions of bodies, whether solid, liquid, or gaseous. 

Classification of Bodies. 

8. Bodies are divided into two classes, solids and fluids. 
A solid is a body which has a tendency to retain a perma- 
nent form. The particles of a solid adhere to each other so 
as to require the action of an extraneous force of greater or 
less intensity to separate them. A fluid is a body whose 
particles move freely amongst each other, each particle yield- 
ing to the slightest force. Fluids are divided into liquids 
and gases, liquids being sensibly incompressible, whilst gases 
are highly compressible. Many bodies are capable of exist- 
ing in either of these states according to their temperature. 
Thus ice, water, and steam, are simply three different states 

of the same body. 

Gravity. 

9. Experiment and observation have shown that the earth 
exercises a force of attraction upon all bodies, tending to 
draw them towards its centre. This force, which is exerted 
upon every particle of every body, is called the force of 
gravity. 

When a body is supported, the force of gravity produces 
pressure or weight ; when it is unsupported, the force pro- 
duces motion. Experiment and observation have shown that 
the entire force of attraction exerted by the earth upon on^ 
body, vanes directly as the quantity of matter in the body, 



16 MECHANICS. 

and i?iversely as the square of its distance from the centre 
of the earth. This force of attraction is mutual, so that the 
body attracts the earth according to the same law. Obser- 
vation has shown that this law of mutual attraction extends 
throughout the universe, and for this reason it has received 
the name of universal gravitation. 

Weight. 

10. The weight of a body is the resultant action of the 
force of gravity upon all of its particles. If the body there- 
fore remain the same, its weight at different places will vary 
directly as the force of gravity, or inversely as the square of 
its distance from the centre of the earth. • 

Mass. 

11. The mass of a body is the quantity of matter which 
it contains. Were the force of gravity the same at every 
point of the earth's surface, the weight of a body might be 
taken as the measure of its mass. But it is found that the 
force of gravity increases slightly in passing from the equa- 
tor towards either pole, and consequently the weight of the 
same body increases as it is moved from the equator towards 
either pole ; its mass, however, remains the same. If we take 
the weight of a body at the equator as the measure of its 
mass, it follows from what has just been said, that the mass 
will be equal to the weight at any place, divided by the force 
of gravity at that place, the force of gravity at the equator 
being regarded as the unit ; or, denoting the mass of any 
body by M, its weight at any place by W, and the force of 
gravity at that place by g, we shall have 

W 

M = — ; whence, W = Mg. 

The expression for the mass of a body is constant, as it 
should be, since the quantity of matter remains the same. 

The unit of mass is any definite mass assumed as a stand- 
ard of comparison. It may be one pound, one ounce, or any 



DEFINITIONS AND INTRODUCTORY REMARKS. 17 

other unit of weight, taken at the equator. The pound ia 
generally assumed as the unit of mass. The terms weight 
and mass may be regarded as synonymous, provided we un- 
derstand that the weight is taken at the equator. 

Density. 

12. The density of a body is the quantity of matter 
contained in a unit of volume of the body, or it is the mass 
of a unit of volume. 

At the same place the densities of two bodies are propor- 
tional to the weights of equal volumes. The mass of any 
body is therefore equal to its volume multiplied by its den- 
sity, or denoting the volume by J 7 ", and the density by D, 
we have 

M = YD. 

We have also, 

M W 

D = ~ = Jl; whence, W = YDg. 

Momentum. 

1 3. The momentum of a moving body, or its quantity 
of motion, is the product obtained by multiplying the mass 
moved, by the velocity with which it is moved ; that is, we 
multiply the number of units in the mass moved by the num- 
ber of units in the velocity with which it is moved and the 
product is the number of units in the momentum. This will 
be explained more in detail hereafter. 

Properties of Bodies. 

14. All bodies are endowed with certain attributes, or 
properties, the most important of which are, magnitude and 
form ' impenetrability ; mobility • inertia y divisibility, and 
porosity ; compressibility, dilatibility and elasticity / at» 
traction, repidsion, and polarity. 

Magnitude and Form. 

15. Magnitude is that property of a body by virtue of 
which it occupies a definite portion of space ; every body 



18 MECHANICS. 

possesses the three attributes of extension, length, breadth, 
and height. The form of a body is its figure or shape. 

Impenetrability. 

16. Impenetrability is that property by virtue of which 
no two bodies can occupy the same space at the same time. 
The particles of one body may be thrust aside by those of 
another, as when a nail is driven into wood ; but where one 
body is, no other body can be. 

Mobility. 

17. Mobility is that property by virtue of which a body 
may be made to occupy different positions at different in- 
stants of time. Since a body cannot occupy two positions 
at the same instant, a certain interval must elapse whilst the 
body is passing from one position to another. Hence motion 
requires time, the idea of time being very closely connected 

with that of motion. 

Inertia. 

18. Inertia is that property by virtue of which a body 
tends to continue in the state of rest or motion in which it 
may be placed, until acted upon by some force. A body at 
rest cannot set itself in motion, nor can a body in motion in- 
crease or diminish its rate, or change the direction of its mo- 
tion. Hence, if a body is at rest, it will remain at rest, or 
if it is in motion, it will continue to move uniformly in a 
straight line, until acted upon by some force. This princi- 
ple is called the law of inertia. It follows immediately 
from this law, that if a force act upon a body in motion, it 
will impart the same velocity, and in the same general di- 
rection as though the body were at rest. It also follows that 
if a body, free to move, be acted upon simultaneously by 
two or more forces in the same, or in different directions, it 
will move in the general direction of each force, as though 
the other did not exist. 

When a force acts upon a body at rest to produce motion, 
or upon a body in motion to change that motion, a resistance 
is developed equal and directly opposed to the effective force 



DEFINITIONS AND INTRODUCTORY REMARKS. 19 

exerted. This resistance, due to inertia, is called the force, 
of inertia. The effect of this resistance is called re-action, 
and the principle just explained may be expressed by saying 
that action and re-action are equal and directly opposed. 
This principle is called the law of action and re-action. 

These two laws are deduced from observation and experi- 
ment, and upon them depends the mathematical theory of 

mechanics. 

Divisibility and Porosity. 

19. Divisibility is that property by virtue of which a 
body may be separated into parts. All bodies may be di- 
vided, and by successive divisions the fragments may be ren- 
dered very small. It is probable that all bodies are composed 
of ultimate atoms which are indivisible and indestructible ; 
if so, they must be exceedingly minute. There are micro- 
scopic beings so small that millions of them do not equal in 
bulk a single grain of sand, and yet these animalcules possess 
organs, blood, and the like. How inconceivably minute, then, 
must be the atoms of which these various parts are composed. 

Porosity is that property by virtue of which the particles 
of a body are more or less separated. The intermediate 
spaces are called pores. When the pores are small, the body 
is said to be dense ; when they are large, it is said to be rare. 
Gold is a dense body, air or steam a rare one. 

Compressibility, Dilatability, and Elasticity. 

20. Compressibility, or contractility, is that property by 
virtue of which the particles of a body are susceptible of 
being brought nearer together, and dilatability is that prop- 
erty by virtue of which they may be separated to a greater 
distance. All bodies contract and expand when their tem- 
peratures are changed. Atmospheric air is an example of 
a body which readily contracts and expands. 

Elasticity is that property by virtue of which a body tends 
to resume its original form after compression, or extension. 
Steel and India rubber are instances of elastic bodies. No 
bodies are perfectly elastic, nor are any perfectly inelastic. 
The force which a body exerts in endeavoring to resume its 



20 MECHANICS. 

form after distortion, is called the force of restitution. If 
we denote the force of distortion by d, the force of restitu- 
tion by r, and their ratio by e, we shall have 

r 

in which e is called the modulus of elasticity. Those 
bodies are most elastic which give the greatest value for e. 
Glass is highly elastic, clay is very inelastic. 

Attraction, Repulsion, and Polarity. 

2 1 . Attraction is that property by virtue of which on e par- 
ticle has a tendency to pull others towards it. Repulsion is 
that property by virtue of which one particle tends to push 
others from it. The dissimilar poles of two magnets attract 
each other, whilst similar poles repel each other. It is sup- 
posed that forces of attraction and repulsion are continually 
exerted between the neighboring particles of bodies, and that 
the positions of these particles are continually changing, as 
these forces vary. 

Polarity is that property by virtue of which the attractive 
and repellent forces between the particles exert an arranging 
power, so as to give definite forms to masses. The phenom- 
ena of crystalization already referred to, depend upon this 
property. It is to polarity that many of the most interest- 
ing phenomena of physics are to be attributed. 

Equilibrium. 

22. A system of forces is said to be in equilibrium when 
they mutually counteract each other's effects. If a system 
of forces in equilibrium be applied to a body, they will not 
change its state with respect to rest or motion ; if the body 
be at rest it will remain so, or if it be in motion, it will con- 
tinue to move uniformly, so far as these forces are concerned. 
The idea of an equilibrium of forces does not imply either 
rest or motion, but simply a continuance in the previous 
state, with respect to rest or motion. Hence two kinds of 
equilibrium are recognized ; the equilibrium of rest, called 



DEFINITIONS AND INTRODUCTORY REMARKS. 21 

statical equilibrium, and the equilibrium of motion, called 
dynamical equilibrium. If we observe that a body remains 
at rest, we infer that all the forces acting upon it are in equi- 
librium ; if we observe that a body moves uniformly, we in 
like manner infer that all the forces acting upon it are in 
equilibrium. 

Definition of Mechanics. 

23. Mechanics is that science which treats of the laws 
of equilibrium and motion. That branch of it which treats 
of the laws of equilibrium is called statics ; that branch 
which treats of the laws of motion is call ed dynamics. When 
the bodies considered are liquids, of which water is a type, 
these two branches are called hydrostatics and hydrodynam- 
ics. When the bodies considered are gases, of which air is 
a type, these branches are called aerostatics and aerody- 
namics. 

Measure of Forces. 

24. We know nothing of the absolute nature of forces, 
and can only judge of them by their effects. We may, how- 
ever, compare these effects, and in so doing, we virtually 
compare the forces themselves. Forces may act to produce 
pressure, or to produce motion. In the former case, they 
are called forces of pressure / in the latter case, moving 
forces. There are two corresponding methods of measuring 
forces, first, by the pressure they can exert, secondly, by the 
quantities of motion which they can communicate. 

A force of pressure may be expressed in pounds; thus, a 
pressure of one pound is a force which, if directed vertically 
upwards, would just sustain a weight of one pound ; a pres- 
sure of two pounds is a force which would sustain a weight 
of two pounds, and so on. 

A moving force may be a single impulse, or it may be 
made up of a succession of impulses. 

The unit of an impulsive force, is an impulse which can 
cause a unit of mass to move over a unit of space in a unit 
of time. A force which can cause two units of mass to move 
over a unit of space in a unit of time, or which can cause a 



22 MECH4NICS. 

unit of mass to move over two units of space in a unit of 
time, is called a double force. 

A force which can cause three units of mass to move over 
a unit of space in a unit of time, or which can cause a unit 
of mass to move over three units of space in a unit of time, 
is called a triple force, and so on. 

If we represent a unit of force by 1, a double force will 
be represented by 2, a triple force by 3, and so on. 

In general, a force which can cause m units of mass to 
move over n units of space in a unit of time, will be repre- 
sented by m x n. Hence, forces may be compared with 
each other as readily as numbers, and by the same general 
rules. 

The unit of mass, the unit of space, and the unit of time, 
are altogether arbitrary, but having been once assumed they 
must remain the same throughout the same discussion. We 
shall assume a mass weighing one pound at the equator, as 
the unit of mass, one foot, as the unit of space, and one 
second, as the unit of time. 

Let us denote any impulsive force, by f the mass moved, 
by m, and the velocity which the impulse can impart to it by 
v. Then, since the velocity is the space passed over in one 
second, we shall have, from what precedes, 

f = mv. 
If we suppose m to be equal to 1, we shall have, 
/= v. 
That is, the measure of an impulse is the velocity which it 
can impart to a unit of mass. 

An incessant force is made of a succession of impulses. It 
has been agreed to take, as the measure of an incessant force, 
the quantity of motion that it can generate in one second, or 
the unit of time. 

If we denote an incessant force by f the mass moved by 
m, and the velocity generated in one second by v, we shal] 
have, 

f — mv. 



DEFINITIONS AND INTRODUCTORY REMARKS. 23 

If we suppose m to be equal to 1, we shall have, 

/=». 

That is, the measure of an incessant force is the velocity 
which it can generate in a unit of mass in a unit of time: 
If the force is of such a nature as to act equally upon 
every particle of a body, as gravity, for instance, the vel- 
ocity generated will be entirely independent of the mass. 
In these cases, the velocity that a force can generate in a unit 
of time, is called the acceleration due to the force. If we 
denote the acceleration by/, the mass acted upon by m, and 
the entire moving force hyf, we shall have, 

f — mf = mv. 

Since an incessant force is made up of a succession of im- 
pulses, its measure may be assimilated to that of an impul- 
sive force, so that both may be represented and treated in 
the same manner. 

Forces of pressure, if not counteracted, would produce 
motion ; and, as they differ in no other respect from the 
forces already considered, they also may be assimilated to 
impulsive forces, and treated in the same manner. 

Representation of Forces. 

25. It has been found convenient in Mechanics to repre- 
sent forces by straight lines; this is readily effected by 
taking lines proportional to the forces which they repre- 
sent. Having assumed some definite straight line to repre- 
sent a unit of force, a double force will be represented by a 
line twice as long, a triple force by a line three times as long, 
and so on. 

A force is completely given when we have its intensity, 
its point of application, and the direction in which it acts. 
When a force is represented by a straight line, the length of 
the line represents the intensity, one 

extremity of the line represents the point • *- 

of application, and the direction of the Fig. 1. 

line represents the direction of the force. 

Thus, in figure 1 , P represents the intensity, the point 



24 



MECHANICS. 



of application, and the direction from to P is the direction 
of the force. This direction is gen- 

erally indicated by an arrow head. jr ^ 

It is to be observed that the point of Fig. i. 

application of a force may be taken 

at any point of its line of direction, and it is often found 

convenient to transfer it from one point to another on this 

line. 

The intensity of a force may be represented analytically 
by a letter, which letter is usually the one placed at the ar- 
row head; thus, in the example just given, we should desig- 
nate the force OP by the single letter P. 

If forces acting in any direction are regarded as positive, 
those acting in a contrary direction must be regarded as nega- 
tive. This convention enables us to apply the ordinary rules 
of analysis to the investigations of Mechanics. 

Forces situated in the same plane are generally referred to 
two rectangular axes, OX and Y, 
which are called co-ordinate axes. 
The direction from towards .Xis 
that of positive abscissas ; that from 
towards X is that of negative ab- 
scissas. The directions from to- ~& 
wards Y and Y\ respectively, are 
those of positive and negative ordi- 
nates. Forces acting in the direc- 
tions of positive abscissas and posi- 
tive ordinates are positive ; those 
acting in contrary directions, are 
negative. 



T' 

Fig. 2. 



Forces in space are referred to 
three rectangular co-ordinate axes, 
OX, Y, and OZ. Forces acting 
from towards X, Y, or Z, are 
positive, those acting in contrary 
directions, are negative. 



/-X' 



yp 



Fii 



. 3. 



COMPOSITION AND RESOLUTION OF FORCES. 25 



CHAPTER II. 

COMPOSITION, RESOLUTION, AND EQUILIBRIUM OF FORCES. 

Composition of Forces whose directions coincide. 

26. Composition of forces, is the operation of finding a 
single force whose effect is equivalent to that of two or more 
given forces. This single force is called the resultant of the 
given forces. Resolution of forces, is the operation of find- 
ing two or more forces whose united effect is equivalent to 
that of a given force. These forces are called components 
of the given force. 

If two forces are applied at the same point, and act in the 
same direction, their resultant is equal to the sum of the two 
forces. If they act in contrary directions, their resultant is 
equal to their difference, and acts in the direction of the 
greater one. In general, if any number of forces are ap- 
plied at the same point, some of which act in one direction, 
and the others in a contrary direction, their resultant is 
equal to the sum of those which act in one direction, dimin- 
ished by that of those which act in the contrary direction • 
or, if we regard the rule for signs, the resultant is equal to 
the algebraic sum of the components ; the sign of this alge- 
braic sum makes known the direction in which the resultant 
acts. This principle follows immediately from the rule 
adopted for measuring forces. 

Thus, if the forces P, P', &c, applied at any point, act in 
the direction of positive abscissas, whilst the forces P", P"\ 
&c, applied to the same point, act in the direction of nega- 
tive abscissas, then will their resultant, denoted by P, be 
given by the equation, 

n = (p + p> + &c.,) - (P" + P'" + ^c.) 

2 



26 MECHANICS. 

If the first term of the second member of this equation is 
numerically greater than the second, R is positive, which 
shows that the resultant acts in the direction of positive ab- 
scissas. If the first term is numerically less than the second, 
R is negative, which shows that the resultant acts in the 
direction of negative abscissas. 

If the two terms of the second member are numerically 
equal, R will reduce to 0. In this case, the forces will exact- 
ly counterbalance each other, and, consequently, will be in 
equilibrium. 

Whenever a system of forces is in equilibrium, their re- 
sultant must necessarily be equal to 0. When all of the 
forces of the system are applied at the same point, this sin- 
gle condition will be sufficient to determine an equilibrium. 

All of the forces of a system which act in the general di- 
rection of the same straight line, are called homologous, and 
their algebraic sum may be expressed by writing the ex- 
pression for a single force, prefixing the symbol 2, a sym- 
bol which indicates the algebraic sum of several homologous 
quantities. We might, for example, write the preceding 
equation under the form, 

X = z{P) (1.) 

This equation expresses the fact, that the resultant of a sys- 
tem of forces, acting in the same direction, is equal to the 
algebraic sum of the forces. 

Parallelogram of Forces. 

27, Let P and Q be two forces applied to the material 
point 0, taken as a unit of mass, and 

acting in the directions OP and OQ. q "R 

Let OP represent the velocity gener- f ^/ 

ated by the force P, and OQ the ve- / ^s^ / 
locity generated by the force Q. Draw L/^ ^J 

PR parallel to OQ, and QR parallel oT~ p 

to OP ; draw also the diagonal OR. Fi Q- 4 

From the law of inertia (Art 18), it follows that a mass 
acted upon by two simultaneous forces moves in the general 



COMPOSITION AND RESOLUTION OF FORCES. 27 

direction of each, as though the other did not exist. Now, 
if we suppose the material point 0, to be acted upon simul- 
taneously by the two forces P and Q, it will, by virtue of the 
first, be found at the end of one second somewhere on the 
line PP ; and by virtue of the second somewhere on the 
line QP ; hence, it will be at their point of intersection. 
But had the point been acted upon by a single force, rep- 
resented in direction and intensity by OP, it would have 
moved from to P in the same time. Hence, the single 
force P is equivalent, in effect, to the aggregate of the two 
forces P and Q ; it is, therefore, their resultant. Hence, 

If two forces be represented in direction and intensity by 
the adjacent sides of a parallelogram, their resultant will be 
represented in direction and intensity by that diagonal of 
the parallelogram which passes through their point of in- 
tersection. 

This principle is called the parallelogram of forces. 

In the preceding demonstration we have only considered 
moving forces, but the principle is equally true for forces of 
pressure ; for, if we suppose a force equal and directly op- 
posed to the resultant P, this force will be in equilibrium 
with the forces P and Q, which will then become forces of 
pressure. The relation between the forces will not be 
changed by this hypothesis, and we may therefore enunciate 
the principle as follows : 

If two pressures be represented in direction and intensity 
by the adjacent sides of a parallelogram, their resxdtant 
will be represented in direction and intensity by that diago- 
nal of the parallelogram which passes through their com 
mon point. 

This principle is called the parallelogram of pressures. 

Hence, we see that moving forces and pressures may be 
compounded and resolved according to the same principles, 
and by the same general laws. 

Parallelopipedon of Forces. 

28. Let P, Q, and S represent three forces applied to 
the same point, and not in the same plane. Upon these lines, 



28 



MECHANICS, 




Fig. 5. 



as ed^cs, -construct the parallelopipedon OP, and draw OM 
and S±L. From the preceding article, 
OM represents the resultant of P and 
Q, and from the same article, OP rep- 
resents the resultant of OM and IS. 

Hence, OP is the resultant of the 
three forces P, Q, and S. That is, if 
three forces be represented in direc- 
tion and intensity by three adjacent 
edges of a parallelopipedon, their residtant will be repre- 
sented by that diagonal of the parallelopipedon which 
passes through their point of intersection. 

This principle is known as the parallelopipedon of forces, 
and is equally true for moving forces and pressures. 

Geometrical Composition and Resolution of Forces. 

29. The following constructions depend upon the prin- 
ciple of the parallelogram of forces. 

1. Having given the directions and intensities of two 
forces applied at the same point, to find the direction and in- 
tensity of their resultant. 

Let OP and OQ represent the 
given forces, and their point of ap- 
plication ; draw PP parallel to Q, 
and QP parallel to OP, and draw 
the diagonal OP ; it will be the re- 
sultant sought. 

2. Having given the direction and intensity of the result- 
ant of two forces, and the direction and intensity of one of 
its components, to find the direction and intensity of the 
other component. 

Let P be the given resultant, P the given component, and 
O their point of application ; drawiLP, and through draw 
OQ parallel to PP, also through P draw PQ parallel to 
P ; then will OQ be the component sought. 

3. Having given the direction and intensity of the result- 
ant of two forces, and the directions of the two components, 
to find the intensities of the components. 




Fig. 6. 



COMPOSITION AND EESOLUTION OF FORCES. 



29 




Let B be the given resultant, OP 
and OQ the directions of the compo- 
nents, and O their point of applica- 
tion. Through B draw BP and BQ 
respectively, parallel to Q and P 0, 

then will OP and Q represent the intensities of the com- 
ponents. 

From this construction it is evident that any force may 
be resolved into two components having any direction what- 
ever; these, again may each be resolved into new compo- 
nents, and so on ; hence it follows that a single force may be 
resolved into any number of components having any as- 
sumed directions whatever. 

4. Having given the direction and intensity of the re- 
sultant of two forces, and the intensities of the components, 
to find their directions. 

Let B be the given resultant, and 
its point of application. With B 
as a centre, and one of the compo- 
nents as a radius, describe an arc of 
a circle ; with as a centre, and the 
other component as a radius, describe 

a second arc cutting the first at P ; draw PB and P 0, and 
complete the parallelogram PQ, then will OP and OQ be 
the directions sought. 

5. To find the resultant of any number of forces, jP, Q, 
JS, T, &c, lying in the same plane, and applied at the same 
point. Construct the resultant B' 
of P and Q, then construct the re- 
sultant B" of B' and £, then the 
resultant B of B" and T, and so on : 
the final resultant will be the result- 
ant of the system. 

By inspecting the preceding fig- 
ure, we see that in the polygon OQ 
B'PJ'BT, the side QB' is equal and 
parallel to the force P, the side 
B'B" to the force &, and the side B"B to the force T % 




Fig. 8. 




SO MECHANICS. 

and so on. Hence, we may construct the resultant of such 
a system of forces by drawing through the second extremity 
of the first force, a line parallel and equal to the second 
force, through the second extremity of this line, a line par- 
allel and equal to the third force, and so on to the last. The 
line drawn from the starting point to the last extremity of 
the last line drawn, will represent the resultant sought. ] f 
the last extremity of the last force fall at the starting point, 
the resultant will be 0, and the system will be in equili- 
brium. 

This principle is called the polygon of forces ; its simplest 
case is the triangle of forces. 

Components of a Force in the direction of two axes. 

30. To find expressions for the components of a force 
which act in directions parallel to two 
rectangular axes. Let OJTarid OYhe 
two such axes, and JR any force lying 
in their plane; construct the compo- 
nents parallel to OX and Y, as be- 
fore explained, and denote the angle 
LAR, which the force makes with the Fi(r 10 

axis of JT, by a. From the figure, we 
have, 

AL = H cos a, and ML = AM = M sin a ; 

or, making A JO = JT, and A M = Y, we have, 

JC = M cos a, and Y = M sin a . . (2.) 

The angle a is estimated from the direction of positive 
abscissas around to the left through 360°. 

For all values of a from 0° to 90°, and from 270° to 360°, 
the cosine of a will be positive, and, consequently, the com- 
ponent AL will be positive ; that is, it will act in the direction 
of positive abscissas. For all values of a from 90° to 270°, 
the cosine of a will be negative, and the component AL 
will act in the direction of negative abscissas. 





J. 


T? 








L 







X 



COMPOSITION AND RESOLUTION OF FORCES. 



3i 




Fig. 10. 



180°, we shall have 



For all values of a from 0° to 180°, the sine of a will be 
positive, and the component AM will 
be positive ; that is, it will act in the 
direction of positive ordinates. For all 
values of a from 180° to 360°, the sine 
of a will be negative, and the compo- 
nent AM will act in the direction of 
negative ordinates. 

For a = 90°, or a — 270°, we shall . 
have AL = 0. For a = 0, or a = 
AM= 0. 

If we regard AL and AM as two given forces, B will be 
their resultant ; and since BL = AM, we shall have from 
the figure, 

B = v / X r TP .... (3.) 

Hence, the resultant of any two forces, at rigid-angles to 
each other, is equal to the square root of the sum of the 
squares of the two forces. 

From the figure, we also have, 

X A . Y 

cos a — — , ana sm a = — • 

Hence, the resultant is completely determined. 

PRACTICAL EXAMPLES. 

1. Two pressures of 9 and 12 pounds, respectively, act 
upon a point, and at right-angles to each other. Required, 
the direction and intensity of the resultant pressure. 



SOLUTION. 



We have, 

X = 9, and T= 12; .-. B 

9 

Also, cos a — — = .6 ; .*. a 

15 

That is, the resultant pressure is 15 lbs., and it makes an 
angle of 53° 7' 47" with the direction of the first force. 
2. Two forces are to each other as 3 is to 4, and their 



'81 + 144 = 15. 
53° 7' 47." 






32 



MECHANICS. 



resultant is 20 lbs. What are the intensities of the compo- 
nents ? 

SOLUTION. 

We have, 3Y = 4X, or Y = f X, and R = 20; 

.-. 20 = y'x 2 + -y 5 -^ 2 = f3T; 

Hence, X - 12, and F" = 16. 

3. A boat fastened by a rope to a point on the shore, is 
urged by the wind perpendicular to the current, with a force 
of 18 pounds, and down the current by a force of 22 pounds. 
What is the tension, or strain, upon the rope, and what 
angle does it make with the current ? 



solution. 



We have 

X = 22, and Y = 18 ; .'. R = y^08 = 28.425 ; 



Also, cos a = 



22 



28.425 



Hence the tension is 28.425 

ir 20". 



■. a — 39° 17' 20". 



5., and the angle 39° 




-<BY 



Components of a Force in the direction of three axes. 
31. To find expressions for the components of a force in 
the directions of three rectangu- 
lar axes. Let OR represent the 
force, and OX, OY, and OZ, 
three rectangular axes drawn 
through its point of application, 
O. Construct a parallelopipedon 
on OR as a diagonal, having 
three of its edges coinciding with 
the axes. Then will the lines 
OZ, OM, and ON, represent 

the required components. Denote these components, re- 
spectively, by X, Y, and Z. Draw lines from R, to L, M, and 



Jl 



fir. n. 



COMPOSITION AND KESOLUTION OF FORCES. 



33 



right-angled 
angle 



N, respectively ; these will be perpendicular to the axes, and 
with them, and the force R, 
will form three 
triangles. Denote the 
between R and the axis of X 
by a, that between R and the 
axis of I^by /3, and that between 
R and the axis of Z by y ; we 
shall have from the right-angled 
triangles referred to, the follow- 
ing equations : 




Fig. 11. 



X — R cos a, Y = R cos /3, and Z — R cos y. 

The angles a, /3, and 7, are estimated from the directions 
of the positive co-ordinates, through 360°. The components 
above found will be positive when they act in the direction 
of positive co-ordinates, and negative when they act in a 
contrary direction. 

If we regard JT, Y, and Z, as three forces, R wdll be 
their resultant, and we shall have, from a known property 
of the rectangular parallelopipedon, 



R = ^X* + Y> + Z* . . . (4.) 

That is, the resultant of three forces at right angles to 
each other^ is equal to the square root of the sum of the 
squares of the components. 

We also have from the figure, 



cos a = -=-, cos p = -=-, and cos y 
Jti Jx 



Z 

~R' 



Hence, the position of the resultant is completely determined. 



EXAM PLES. 



1. Required the intensity and direction of the resultant 
of three fo^es at right angles to each ?ther, having the in- 
tensities 4, 5, and 6 pounds, respectively. 
2* 



'64 MECHANICS. 



SOLUTION 

"W e have, 
X = 4, T = 5, and Z = 6. .\ i? = v/77= 8.775. 

Also, cos* = ^/co S ^ = ^,andcos 7 = ^: 
whence, a = 62°52'51", = 55°15'50", and 7 = 46°51'48". 

Hence the resultant pressure is 8.775 lbs., and it makes, with 
the components taken in order, angles equal to 62° 52' 51", 
55° 15' 50", and 46° 51' 43". 

2. Three forces at right angles are to each other as the 
numbers 2, 3, and 4, and their resultant is 60 lbs. What are 
the intensities of the forces ? 



SOLUTION. 

We have 

Y = fX, Z =' 2X, and R = 60 ; 
Tlence, 

60 = ^/X 2 + f X 2 + 4X 2 = iJ^9 = 2.6925X. 
.'. X = 22.284. 
The components are, therefore, 

22.284 lbs., 33.426 lbs., and 44.568 lbs. 
Projection of Forces. 

32. If planes be passed through the extremities of a 
force, perpendicular to the direction of any straight line, 
that portion of the line intercepted between them is the pro- 
jection of the force upon the line. The operation of resolv- 
ing forces into components in the direction of rectangular 
axes, is nothing more than that of finding their projections 
upon these axes. 

If two straight lines be drawn through the extremities of 
a force, perpendicular to any plane, and the points in which 
they meet the plane be joined by a straight line, this line is 
the projection of the force upon the plane. 



COMPOSITION AND RESOLUTION OF FORCES. 35 

If we denote any force by P, and the angle which it 
makes with any line or plane by a, P cos a will represent 
the projection of the force on the line or plane. In both 
cases the projection of the force is its effective component in 
the direction of the line or plane npon which it is projected. 

Composition of a Group of Forces in a Plane. 

33. Let P, P', P", &c, denote any number of forces 
lying in the same plane, and applied at a common point, and 
represent the angles which they make with the axis of X by 
«, a', a", &c. Their components hi the direction of the axis 
^f X are P cos a, P' cos a', P" cos a", <fcc, and their com- 
ponents in the direction of the axis of Y, are P sin a, 
P' sin a', P" sin a", &c. 

If we denote the resultant of the group of components 
which are parallel to the axis of X by AT, and the resultant 
of the group parallel to the axis of Y by Y, we shall have, 
(Art. 26), 

X— 2 (P cos a), and Y = 2 (P sin a) . . (5.) 

The resultant of AT and I^is the same as the resultant of the 
given forces. Denoting this resultant by P, and recollecting 
that X and Y are perpendicular to each other, we have, as 
in Article 30, 

p = jx* + r* . . . . ( 6.) 

if we denote the angle which the resultant makes with the 
avis of X by a, we shall have, as in Article 30, 

X ' . Y 

cos a = -g r , and sm a — -— • 
-Li Jx 

EXAMPLES. 

1. Three forces, whose intensities are respectively equal 
to 50, 40, and 70, lie hi the same plane, and are applied at 
the same point, and make with an axis through that point, 
angles equal to 15°, 30°, and 45°, respectively. Required 
the intensity and direction of the resultant. 



36 



MECHANICS. 



S OLUTION. 

We have, 

X = 50 cos 15° + 40 cos 30° + 70 cos 45° = 132.435, 
and 

T= 50 sin 15° + 40 sin 30° + 70 sin 45° = 32.44 ; 
whence, 

R = -y/6798 -f- 17539 = 156. 

and cos a = ' — ; .-. a = 31° 54' 12". 

156 ' 

The resultant is 156, and the angle which it makes with the 
axis is equal to 31° 54' 12". 

2. Three forces 4, 5, and 6, lie in the same plane, making 
equal angles with each other. Required the intensity of 
their resultant and the angle which it makes with the least 
force. 

SOLUTION. 

Take the least force as the axis of X. Then the angle 
between it and the. second force is 120°, and that between it 
and the third force is 240°. We have 

X = 4 ' + 5 cos 120° 4- 6 cos 240° = — 1.5 ; 
T= 5 sin 120° + 6 sin 240° = — .866; 

+, r- 1.5 .866 

•*• -** = V 3, cos a = — — -— ,sma = — -~— — ; 
v ' 1.732' 1.732' 

.-. a= 210°. 

3. Two forces, one of 5 lbs. and the other of 7 lbs., are 
applied at the same point, and make with each other an 
angle of 12©°. What is the intensity of their resultant? 

Ans. 6.24 lbs. 

Composition of a Group of Forces in Space. 
34. Let the forces be represented by JP, P', P", &c. 

The angles which they make with the axis of X, by a, a', a", 
<fca, the angles which they make with the axis of Y, by /3, 
(3\ /3", &c, and the angles which they make with the axis' 



COMPOSITION AND RESOLUTION OF FORCES, 37 

of Z by 7, 7', y", &c. Resolving each force into compo- 
nents, respectively parallel to the three co-ordinate axes, 
and denoting the resultants of the groups in the directions 
of the respective axes by -X", Y, and Z, we shall have, as in 
the preceding article, 

X = 2 (P cos a), T= 2 (P cos (3), Z = 2 (P cos 7.) 

If we denote the resultant of the system by P, and the 
angles which it makes with the axes by a, b, and c, we shall 
have, as in Article 31, 



r = y'x 2 + r a + z\ 

X Y Z 

cos a= -5, cos b = -5, and cos c — — • 
H Jx Jti 

The application of these formulas is entirely analogous to 
that of the formulas in the preceding article. 

Expression for the Resultant of two Forces. 

35. Let us consider two forces, P and P', situated in 
the same plane. Since the position 
of the co-ordinate axes is perfectly 
arbitrary, let the axis of X be so 
taken as to coincide with the force 
P; a, will then be equal to 0, and we 
shall have sin a = 0, and cos a = 1 . Fig . i2 , 

The value of X (Equation 5), will 

become P + P' cos a', and the value of Y will be- 
come P' sin a'. Squaring these values, substituting 
them in Equation ( 6 ), and reducing by the relation 
sin a a! -f- cos 2 a! = 1, we have, 



B = t/P* + P' 8 + 2PP' cos a' . ( 7.) 

The angle a' is the angle included between the given 
forces. Hence, 

The resultant of any tico forces, applied at the same 
point, is equal to the square root of the sum of the squares 



38 



MECHANICS. 



of the two forces, plus twice the product of the forces into 
the cosine of their included angle. 

If we make a' greater than 90°, and less than 270°, its 
cosine will be negative, and we shall have, 



R = yTM- P" 2 - 2PP' cos a'. 

If we make a' — 0, its cosine will be 1, and we shall 
have, 

R = P + P\ 

If we make a' = 90°, its cosine will be equal to 0, and 
we shall have, 

R = -y/F* +P'\ 

180°, its cosine will be — 1, and we 
R = P - P'. 



If we make a' 
shall have, 



The last three results conform to principles already de- 
duced. Let P and Q be two forces, 
and P their resultant. The figure 
QP being a parallelogram, the 
side PR is equal to Q. From the 
triangle OPP we have, in accor- 
dance with the principles of trigo- 
nometry, 




Fig. 13. 



P : Q : P : : sin OPP : sin POP : sin OPP. ( 8.) 

If we apply a force P' equal and directly opposed to R, 
the forces P, Q, and R\ will be in 
equilibrium. The angles ORP, 
and QOR\ being opposite exte- 
rior and interior angles, are sup- 
plements of each other ; hence, 
sin ORP = sin QOR'. The 
angles R OP, and P OR', are ad- 
jacent, and, consequently, supple- 
mentary ; hence, sin POP = sin POP'. The angles 





COMPOSITION AND RESOLUTION OF FORCES. 3D 

OPR, and POQ, are interior 
angles on the same side, and, con- 
sequently, supplementary ; hence, 
sin OPR - sin POQ. We have 
also R — R'. Making these sub- 
stitutions in the preceding propor- 
tion, we have, Fi s u - 

P : Q : R' : : sin QOR' : sin POR' : am POQ. 

Hence, if three forces are in equilibrium, each is propor- 
tional to the sine of the angle between the other two. 

EXAMPLES. 

1. Two forces, P and Q, are equal in intensity to 24 and 
30, respectively, and the angle between them is 105°. What 
is the intensity of their resultant ? 



R = V24 2 + 30 2 + 2 X 24 X 30 cos 105° = 33.21. 

2. Two forces, P and Q, whose intensities are respec- 
tively equal to 5 and 12, have a resultant whose intensity is 
13. Required the angle between them. 

13 = y/2b -f 144 + 2 X 5 X 12 cos a. 

.'. cos a = 0, or a = 90°. Ans. 

3. A boat is impelled by the current at the rate of 4 
miles per hour, and by the wind at the rate of V miles per 
hour. What will be her rate per hour when the direction 
of the wind makes an angle of 45° with that of the current ? 

R = i/l6 + 49 + 2 x 4 x 1 cos 45° = 10.2m. Ans. 

4. A weight of 50 lbs., suspended by a string, is drawn 
aside by a horizontal force until the string makes an angle 
of 30° with the vertical. Required the value of the hori- 
Eontal force, and the tension of the string. 

Ans. 28.8675 lbs., and 51.135 lbs. 



40 MECHANICS. 

5. Two forces, and their resultant, are all equal. What 
is the value of the angle between the two forces ? 120°. 

6. A point is kept at rest by three forces of 6, 8, and 11 
lbs., respectively. Required the angles which they make 
with each other. 

SOLUTION. 

We have P = 8, Q — 6, and B! = 11. Since the 
forces are in equilibrium, we shall have B! = H =11; 
hence from the preceding article, 



11 = V64 + 36 + 96 cos QOP; 
.\ cos QOP = |f; or, QOP = 11° 21' 52". 

From the last proportion we have, 

sin POP' 6 • ' »Vit>, trtrtrtJ1 

sin (tfl/.r' 11 

or r POP' = 147° 50' 34". 

or, QOP' = 134° 47' 34" 

Principle of Moments. 

36. The moment of a force, with respect to a point, is 
the product obtained by multiplying the intensity of the 
force by the perpendicular distance from the point to the 
line of direction of the force. 

The fixed point is called the centre of moments ; the per- 
pendicular distance is called the lever arm of the force ; and 
the moment itself measures the tendency of the force to 
produce rotation about the centre of moments. 



COMPOSITION AND RESOLUTION OF FORCES. 



41 




Let P and Q be any two 
forces, and B their resultant ; 
assume any point C, in then- 
plane, as the centre of moments, 
and from it, let fall upon the di- 
rections of the forces, the per- 
pendiculars, Cp, Cq, and Cr; 
denote these perpendiculars res- 
pectively by p, q, and r. Then will Pp, Qq, and Br, be 
the moments of the forces P, Q, and B. Draw CO, and 
from P let fall the perpendicular PS, upon OB. Denote 
the angle B OP, by a, the angle BOQ, or its equal, OBP, 
by (3, and the angle BOCbj 9. 

Since PB = Q, we have from the right-angled triangles 
OPS and PBS, the equations, 

B = Q cos (3 + P cos a. 

= Q sin (3 — P sin a. 

Multiplying both members of the first equation by sin 9, 
and both members of the second by 00s 9, then adding the 
resulting equations, we find, 

B sin cp = Q (sin 9 cos (3 4- sin j3 cos 9) + 
P (sin 9 cos a — sin a cos 9). 

Whence, by reduction, 

B sin 9 = Q sin (9 -f /3) + P sin (9 — a). 



From the figure, we have, 



Bin 9 



-oc* sin (? 



P 
OC' 



and sin (9 + (3) 



00' 



Substituting in the preceding equation, and reducing, we 
have, 

Br=Qq + Pp. 

When the point G falls within the angle P OB, 9 — a 
becomes negative, and the equation just deduced becomes 



Br = Qq - Pp. 



42 MECHANICS. 

Hence, we conclude in all cases, that the moment of the 
resultant of two forces is equal to the algebraic sum of the 
moments of the forces taken separately. 

If we regard the force Q as the resultant of two others, 
and one of these in turn, as the resultant of two others, and 
so on, the principle may be extended to any number of 
forces lying in the same plane, and applied at the same 
point. This principle may, in the general case, be expressed 
by the equation 

Br = 2 (Pp) (9.) 

That is, the moment of the resultant of any number of 
forces, lying in the same plane, and applied at the sam.e 
point, is equal to the algebraic sum of the moments of the 
forces taken separately. 

This is called the principle of moments. 

The moment of the resultant is called the resultant mo- 
ment ; the moments of the components are called compo- 
nent moments ; and the plane passing through the resultant 
and centre of moments, is the plane of moments. 

When a force tends to turn its point of application about 
the centre of moments, in the direction of the motion of 
the hands of a watch, its moment is considered positive ; 
consequently, when it tends to produce rotation in a contrary 
direction, the moment must be negative. If the resultant 
moment is negative, the tendency oi the system is to pro- 
duce rotation in a negative direction about the centre of 
moments. If the resultant moment is 0, there is no ten- 
dency to produce rotation in the system. The resultant 
moment may become 0, either in consequence of the lever 
arm becoming 0, or in consequence of the resultant itself 
being equal to 0. In the former case, the centre of mo- 
ments lies upon the direction of the resultant, and the nu- 
merical value of the sum of the moments of the forces 
which tend to produce rotation in one direction, is equal to 
that of those which tend to produce motion in a contrary 
direction. In the latter case, the system of forces is li 
equilibrium. 




COMPOSITION AND RESOLUTION OF FORCES. 43 

Moments, with respect to an Axis. 

37. To form an idea of the moment of a force with 
respect to a straight line, taken 

as an axis of moments. Let P 
represent any force, and let the 
axis of Z be assumed so as to 
coincide with the axis of mo- 
ments. Draw the straight line 
AB perpendicular, both to the 
direction of the force and to the 
axis of moments ; at the point Fig. 16. 

A, in which this perpendicular 

intersects the direction of the force, let the force P be 
resolved into two components, P" and P', the first parallel 
to the axis of Z, and the second at right angles to it. The 
former will have no tendency to produce rotation, the latter 
will tend to produce rotation, which tendency will be mea- 
sured by P' x AB; this product is the moment of the 
force P with respect to the axis of moments, and is evi- 
dently equal to the moment of the projection of the force 
upon a plane at right-angles to the axis, taken with respect 
to the point in which this axis pierces the plane as a centre 
of moments. 

If there are any number of forces situated in any manner 
in space, it is clear from the preceding principles that their 
resultant moment, with respect to any straight line taken as 
an axis of moments, is equal to the algebraic sum of the 
component moments with respect to the same axis. 
Principle of Virtual Moments. 

38. Let P represent a force applied to the material 
point ; let the point be moved by 

an extraneous force to some position, p' Op JP 

C, very near to ; project the path £'' N g 

upon the direction of the force ; rig> 17 . 

the projection Op, or Op', is called the 

virtual velocity of the force, and is taken positively when 

it falls upon the direction of the force, as Op, and nega- 



44 



MECHANICS. 




Fig. 18. 



lively when it falls upon the prolongation of the force, as 
Op' . The product obtained by multiplying any force by its 
virtual velocity is called the virtual moment of the force 

Assume the figure and nota- 
tion of Article 36. Op, Oq, and 
Or are the virtual velocities of 
the forces P, Q, and P. Let 
us denote the virtual velocity of o< 
any force by the symbol of va- 
riation 8, followed by a small 
letter of the same name as that 
which designates the force. 

We have from the figure, as in Article 36, the relations, 

P — P cos a + Q cos (3. 

= P sin a — Q sin (3. 

Multiplying both members of the first by cos <p, and of the 
second by sin 9, and adding the resultant equations, we have, 

P cos 9 = P (cos a cos 9 -f sin a sin 9) -4 
Q (cos 9 cos (3 — sin 9 sin (3). 

Or, by reduction, 

P cos 9 = P cos (9 — a) + Q cos (9 + (3). 

But, from the right-angled triangles C Op, OOq, and O Or, 
we have, 



8r 



§p 



COS9 = T r ?y , cos (9 - a) = _^, and cos (9 + /3) = y 



00 



oc 



00' 



Substituting these in the preceding equation, and reducing, 
we have, 

PSr = PSp + QSq. 

Hence, the virtual moment of the resultant of two forces, 
is equal to the algebraic sum of the virtual moments of the 
two forces taken separately. 



COMPOSITION AND RESOLUTION OF FORCES. 45 

If we regard the force Q as the resultant of two other 
forces, and one of these as the resultant of two others, and 
so on, the principle maybe extended to any number of forces, 
applied at the same point. This principle may be expressed 
by the following equation : 

Hdr = 2 (PSp) .... (10.) 

Hence, the virtual moment of the resultant of any num- 
ber of forces applied at the same point, is equal to the alge- 
braic sum of the virtual moments of the forces taken sepa- 
rately. 

This is called the principle of virtual moments. If the 
resultant is equal to 0, the system is in equilibrium, and the 
algebraic sum of the virtual moments is equal to ; con- 
versely, if the algebraic sum of the virtual moments of the 
forces is equal to 0, the resultant is also equal to 0, and the 
forces are in equilibrium. 

This principle, and the preceding one, are much used in 
discussing the subject of machines. 

Resultant of parallel Forces. 
39. Let P and Q be two forces lying in the same plane, 
and applied at points invariably 
connected, for example, at the 
points M and N of a solid body. 
Their lines of direction being pro- 
longed, will meet at some point ^,-rll 
; and if we suppose the points 
of application to be transferred to 
0, their resultant may be deter- 
mined by the parallelogram of forces. The direction of the 
resultant will pass through 0. (Art. 27.) Whether the 
forces be transferred to or not, the direction of the resul- 
tant will always pass through 0, and this whatever may be 
the value of the included angle. Now, supposing the points 
of application to be at M and jV, let the force Q be turned 
about N~ as an axis. As it approaches parallelism with P, 





46 MECHANICS. 

the point will recede from M and JST, and the resultant will 
also approach parallelism with P. 
Finally, when Q becomes parallel 
to P, the point O will be at an 
infinite distance from M and JV, 
and the resultant loill also be par- 
allel to P and Q. In any position °°^ ' ^£ 
of P and Q, the value of the re- Fig. 19. 

sultant, denoted by B, will be 
given by the equation (Art. 36), 

P =. Pcbsz -+- §cos/3. 

When the forces are parallel, and lying in the same direc- 
tion, we shall have a = 0, and ft = 0; or, cos a = 1, and 
cos (3 = 1. Hence, 

B = P ■+ Q. 

If the forces lie in opposite directions, we shall have 
a = 0, and (3 = 180°; or, 
cos a = 1, and cos /3 = — 1. 
Hence, .'''' X s 

P = P- Q. 




0-- 



That is, the resultant of two 
parallel forces is equal in inten- 
sity to the algebraic sum of the "~W 
forces, and its line of direction Fig 20. 
is parallel to that of the two forces. 

If we regard Q as the resultant of two parallel forces, and 
one of these as the resultant of two others, and so on, the 
principle may be extended to any number of parallel forces. 
Denoting the resultant of a group of parallel forces, 
P, P\ P", &c, by B, we have, 

* = *(P) (11.; 

That is, the resultant of a group of parallel forces is 
equal in intensity to the algebraic sum of the forces. Ps 
line of direction is also parallel to that of the given forces. 



COMPOSITION AND RESOLUTION OF FORCES. 47 

Point of Application of the Resultant. 
40. Let P and Q be two parallel forces, and P their 
resultant. Let M and N be 

the points of application of -^ ^ 

the two forces, and S the 

point in which the direction f,^~ 

of R cuts the line MN. MI^ 

Through JV draw NL per- Fi 21 

pendicular to the general di- 
rection of the forces, and assume the point (7, in which it 
intersects the line of direc- 
tion of P, as a centre of mo- „ N" 

Q -* * ~p 

ments. Since the centre of 



-U 



L 



-^P 



5-H 



c 

Fig. 22. 



moments is on the line of / 

direction of the resultant, S6 1 — 

the lever arm of the resultant 

will be 0, and we shall have, 

from the principle of moments (Art. 36), 

Px GZ= Q x CAT; 

or, P : Q : : CJV : CL. 

But, from the similar triangles CNS and LNM, we ha\ e, 

CN : CL : : SN : SM. 

Combining the two proportions, we have, 

P : Q : : SN : SM. 

That is, the line of direction of the resultant divides the 
line joining the points of application of the components, 
inversely as the components. 

From the last proportion, we have, by composition, 

P : Q : P + Q : : SJST : SM : SJST + SM; 
and, by division, 

P : Q : P- Q : : SN- SM : SN- SM. 



48 



MECHANICS. 



When the forces act in the same direction, P + Q will 
be their resultant, and SN -\- SM will equal MN. Since 
P -f- Q is greater than either P or Q, JOT will be greater 
than either SN or SM, which shows that the resultant lies 
between the components. 

When the forces act in contrary directions, P — Q will 
be their resultant, and SN — SM will equal MN Since 
P — Q is less than P (supposed the greater of the compo- 
nents), MN will be less than SN, which shows that the 
resultant lies without both components, and on the side of 
the greater. 

Substituting in the preceding proportions, for P + Q, 
P— Q, SN + SM, and SN — SM, their values, we have, 

P : Q : B : : SN : SM : MN. . . . (8)', 

That is, of two parallel forces and their resultant, each is 
proportional to the distance between the other two. 

Geometrical Composition and Resolution of Parallel Forces. 
41. The preceding principles give rise to the following 



geometrical constructions : 

1. To find the resultant of two 
parallel forces lying in the same direc- 
tion: 

Let P and Q be the forces, M and 
N their points of application. Make 
MQ' = ft and NP' = P; draw P' Q', 
cutting MN in S ; through S draw SP 
parallel to MP, and make it equal to 
P+Q: it will be the resultant. 

For, from the similar triangles P'SN 
and Q'SM, we have, 



/?' 



*-f B — 



r J 



Fig. 23. 



PN : Q'M : : SN : SM; or, P : Q : : SN : SM 

After the construction is made, the distances MS and 
NS may be measured by a scale of equal parts. 



COMPOSITION AND RESOLUTION OF FORCES. 49 

EXAMPLE. 



Given P = 9 lbs, Q 
quired MS. 

We have P = 15, hence, 

15 : 6 : : 30 : MS; 



6 lbs., and MN = 30 in. Re- 



MS = 12 in. Ans. 



M S 



2. To find the resultant of two parallel forces acting in 
opposite directions : 

Let P and Q be the forces, M and 
N their points of application. Prolong 
QNtittNA = P, and make MB = Q; 
draw AB, and produce it till it cuts 
NM produced in S ; draw SR parallel 
to J/P, and make it equal to BP, it 
will be the resultant required. 

For from the similar triangles SNA 
and SMB, we have, 



~A 



/R 



AN : BM : : SN : SM; or, P : § 



Fig. 24. 

: SN : $$L 



EXAMPLE. 

Given P = 20 lbs., § = 8 lbs., and NM = 18 in. 
Required SN 

We have 22 = 20 — 8 = 12; hence, from Proportion (8) J 

12 : 20 : : 18 : SN; .'. SN = 30 in. ^s. 

3. To resolve a given force into two parallel components 
lying in the same direction, and applied at given points : 

Let P be the given force, M and 
N the given points of application. 
Through M and N draw lines parallel 
to P. Make MA = P, and draw AN, 
cutting P in B ; make ilfP = #2? and 
A r § — BR; they will be the reqv; d 
components. 
3 



Al-' 



X 



Fit:. 25 



50 MECHANICS. 

For, from the similar triangles AMN and BSN, 
BS : AM : : SN : MN; 



or, BS : B : : SN : MN 
But, from Proportion (8)', we have, 
P : B : : SN : MN; 
.\ BS --= P, and BB = Q. 









• — 7B 



Rt 



/ 



Fig. 25. 



;q 



EXAMPLE. 

24 lbs., £Jf = 7 in., and SN = 5 in. Re. 



Given B 
quired P and Q. 

From Proportion (8), we have, 

12 : 7 : : 24 : Q ; . 
12 : 5 : : 24 : P ; 



Q = 14 lbs. 
P = 10 lbs. 



4. To resolve a given force into parallel components lying 
in opposite directions, and applied at given points. Both 
points of application must lie on the same side of the given 
force. Let B be the given force, M and N the given 
points of application. Through M 
and iVdraw lines parallel to B ; make L„ A 

NB = B, and draw BM; through 
S, draw SA parallel to MB', then 
will NA and BA be equal to the in- 
tensities of the components. Make 
MP = AN, and NQ = AB, and 
they will be the components. For, 
from the triangles ASN, and BMN, 
we have, 






C_jB 



l£""°N 



: : i 



AN : BN : : SN: MN\ or, ^liV : B : : SN : 3IN 
But, from Proportion (8)', we have, 
P : B : : SN : MN; /. AN = P, and ^4P = (>. 



COMPOSITION AND RESOLUTION OF FORCES. 



51 



r 



tfr|, 



IT 



EXAMPLE. 

Given R = 24 lbs., j82V = is in., and SM = 9 in. Re. 
quired P and §. 

From Proportion (8)', we have, 

P : 24 : : 18 : 9 ; .\ P = 48 lbs. 
§ : 24 : : 9:9; /. § = 24 lbs. 

P = P - Q = 24 lbs. 

5. To find the resultant of any number of parallel forces. 

Let P, P', P", P'", be such a system of forces. Find 
the resultant of P and P', by the rule 
already given, it will be R' = P -f P' ; 
find the resultant of P' and P', 
it wiU be R" = P + P + P"; find 
the resultant of R" and P", it will be 
R = P + P + P" + P" If there 
is a greater number of forces, the 
operation of composition may be con- 
tinued ; the final result will be the re- 
sultant of the system. If some of the 
forces act in contrary directions, combine all which act in 
one direction, as just explained, and call their resultant R' ; 
then combine all those which act in a contrary direction, 
and call their resultant R" ; finally, combine R' and R" by 
a preceding rule ; their resultant R will be the resultant of 
the system. 

If R' — R'\ the resultant will be 0, and its point of ap- 
plication will be at an infinite distance. In this case, the 
forces reduce to a couple, the effect of which is simply to pro- 
duce rotation. 

Lever Arm of the Resultant. 

42. Let P, P', P", &c, denote any number of parallel 
forces, and p, p\ p'\ &c, their lever arms with respect to an 
axis of moments, taken perpendicular to the common direc- 
tion of the forces ; denote the lever arm of the resultant of 



Fig. 27. 



52 



MECHANICS. 



the system, taken with respect to the same axis, by r From 
the principle of moments (Art. 37), 

(P + P + P" + &e.)r = Pp + P'p' + &c. ; 

-|^ • • • (-.) 

Hence, £Ae fever arm o/* the resultant of a system of par- 
allel forces, with respect to an axis at right-angles to their 
direction, is equal to the algebraic sum of the moments of 
the forces divided by the algebraic sum of the forces. 

Centre of Parallel Forces. 
43. Let there be any number of forces, P, P', P ', &c, 

applied at points invariably connected together, and whose 
co-ordinates are x, y, z; x', y\ z ; jc", y", z" ; &c. Let R 
denote their resultant, and represent the co-ordinates of its 
point of application, by x x , y 15 and z Y ; denote the angles made 
by the common direction of the forces with the axes of 
-3T, Y", and Z, by a, /3, and y. 

Suppose each force resolved into three components, re- 
spectively parallel to the co-ordinate axes, the points of 
application being unchanged : 

The components parallel to the axis of _3T are, 

Pcosa, P'cosa, P"cosoi, &c, Pcosa ; 
those parallel to the axis of 3^ are, 

Pcos/3, P'cos,3, P"cos/3, <fcc, Pcos/3 ; 
ind those parallel to the axis of Z are, 

Pcos/, P'cos/, P"cos/, &c, Pcos/. 

If we take the moments of the components parallel to the 
sxis of Z, with respect to the axis of IT, as an axis of mo- 
aaetts, we shall have, for the lever arms of the components, 
r, x\ x", &c. ; and from the principle of moments (Art. 37), 

Pcos/ a?j = Pcosy x -f P'cos;- x' -f- &c. 



COMPOSITION AND RESOLUTION OF FORCES. 53 

Striking out the common factor cos 7, and substituting 
for B, its value, we have, 

whence, x l = -±JQ • 

In like manner, if % we take the moments of the same com- 
ponents, with respect to the axis of J£T, we shall have, 

v - *( p y) 

yi ~ 2(P) 

And, if we take the moments of the components parallel 
to the axis of YJ with respect to the axis of JT, we shall 
have, 

2(A) 



«i = 



2(P)- 



Hence we have for the co-ordinates of the point of appli- 
cation of the resultant, 

*i - 2(p) , 2/i - 2(p) , and 0, _ 2(p) . (13.) 

These co-ordinates are entirely independent of the direc- 
tion of the parallel forces, and will remain the same so long 
as their intensities and points of application remain un- 
changed. 

The point whose co-ordinates we have just found, is called 
the centre of parallel forces. 

Resultant of a Group of Forces in a Plane, and applied at points 
invariably connected. 

44. Let P, P, P", &c, be any number of forces lying 
in the same plane, and applied at points invariably connected 
together ; that is, at points of the same solid body. 



a. -. 



X 

Fig. 28. 



54 MECHANICS. 

Through any point in the plane of the forces, draw any 
two straight lines, X and Y, 
at right angles to each other, and 
lying in the plane of the forces ; 
assume these as co-ordinate axes. 
Denote the angles which the 
forces P, P', P', &c, make with — 
the axis OAT, by a, a', a", <fcc, 
and the angles which they make 
w T ith the axis OY> by (3, /3', /3", &o. ; denote, also, the co- 
ordinates of the points of application of the forces, by a?, y ; 

*'> y' ; ®"> y" ; &c - 

Let each force be resolved into, components parallel to the 
co-ordinate axes ; we shall have for the group parallel to the 
axis of AT, 

Pcosx, jFcosa', P'cosx", &c. ; 

and, for the group parallel to the axis of Y, 

Pcos,3, Pcos/3', P"cos/3", <fcc; 

The resultant of the first group is equal to the algebraic 
sum of the components (Art. 39) ; denoting this by JT, we 
shall have, 

AT=2(Pcosa) .... (14.) 

In like manner, denoting the resultant of the second group 
by Y, we shall have, 

F = 2(Pcos/3) .... (15.) 

The forces X and Y intersect in a point, which is the 
point of application of the system of forces. Denoting the 
resultant by P, we shall have (Art. 33), 



B = /T+ Y\ 

To find the point of application of P, let be taken as a 
centre of moments, and denote the lever arms of AT and Y 



COMPOSITION AND RESOLUTION OF FORCES. 55 

by y l and x^ respectively. From the principle of Article 
42, we shall have, 

Xl ~ 2(Pcos,tf) ' * " * l b ° 

• S(Pcosay) 
2/1 - 2(Pcosa) • • ' • U7.) 

Il we denote the angles which the resultant makes with 
the axes of X. and Y by a and b respectively, Ave shall 
have, as in Article 33, 

-5T Y 

cos a =^5, cos b = -= . . . (18.) 

Equations (16) and (17) make known the point of applica- 
tion, and Equations (18) make known its direction ; hence, 
the resultant is completely determined. 

To find the moment of P, with respect to as a centre 
of moments, let us denote its lever arm by r, and the lever 
arms of P, P, P", &c, with respect to 0, hyp,p\p", &c. 

The moment of the force Pcosa, is Pcosxy, and that 
of the force Pcos3, is — Pcos/3 x. The negative sign is 
given to the last result, because the forces Poos?, and 
PcosS tend to turn the system in contrary directions. 

From the principle of moments (Art. 36), the moment of 
P is equal to the algebraic sum of the moments of its com- 
ponents. Hence, 

Pp = Pcosa y — Pcos/3 x. 

In like manner, the moments of the other component 
forces may be found. Because the moment of the resultant 
is equal to the algebraic sum of the moments of all its com- 
ponents (Art. 36), we have, 

Rr = 2(Pp) = 2 (Pcosa y - Pcos/3 x) . (19.) 



56 



MECHANICS. 




T-'^Pcosa 



Fig. 29. 



Resultant of a Group of Forces situated in Space, and applied at 
points invariably connected. 

45. Let P, P', P", &c, be any number of forces 
situated in any manner in space, 
and applied at points of the ^ 

same solid body. Assume any » 

point in space, and through 
it draw any three lines perpen- 
dicular to each other. Assume 
these lines as axes. Denote the 
angles which the forces P, P', 
P', &c, make with the axis of 
JX, by a, a', a",'<fcc. ; the angles 
which they make with the axis 
of Y, by /•?, /3', /3", <fcc. ; the angles which they make with 
the axis of Z, by /, y\ y" r &c, and denote the co-ordi- 
nates of their points of. application by se, y, s; a*', y', s' ; 

sb". y", *"; &G - 

Let each force be resolved into components respectively 
parallel to the co-ordinate axes. 

We shall have for the group parallel to the axis of _5T, 

Pcosa, Pcosa', P'cosx", &c. ; 
for the group parallel to the axis of Y, 

Pcos/3, Pcos/3', P'cos/3", Ac. ; 
and for the group parallel to the axis of Z, 

Pcos/, Pcosy', P'cos/", &c. 

Denoting the resultants of these several groups by JT, Y, 
and Z, we shall have, 

X=2(Pcosa,) F= 2 (Pcos/3,) and Z = 2 (Pcosy) . (20.) 

If these three forces intersect at a point, this point is ' 
the point of application of the resultant of the entire ays- 



COMPOSITION AND RESOLUTION OF FORCES. 57 

tern. Denote this resultant by B ; then, since the forces 
JT, Y, and Z are perpendicular to each other, we shall have, 



R = t/X 2 + Y* + Z a . . . (21.) 

To find the Co-ordinates of the point of application of R. 

Consider each of the forces, .X, Y, and Z, with respect to 
the axis whose name conies next in order, and denote the 
lever arm of JT, with respect to the axis of Y, by z x ; that 
of Y", with respect to the axis of Z, by x 1 ; and that of Z, 
with respect to the axis of X, by y Y . We shall have as in 
the last article, 

^(Pcos ffa:) 

Xi ~ ~Y(p^o^y 



(22.) 



S(Pcosyy) 
yi 2(Pcos/) 

2(Pcosa z) 
Zl = 2(Pcosa) 



in which «„ y^ and s 1? are the co-ordinates of the point of 
application of i?. 

Denoting the angles which JR makes with the axes by 
a, 5, and c, respectively, we have, as in the preceding 
article, 

X T Y Z , x 

cos a = -^3, cos = -= , cos c = -^ . . (23.) 
Jti Ji Ji 

The values of X, Y, and Z, may be computed by means of 
Equations (20), and these being substituted in (21), make 
known the value of the resultant. The co-ordinates of 
its point of application result from Equations (22), and its 
line of direction is shown by Equations (23). The intensity, 
direction, and point of application being known, the resul- 
tant is completely determined. 



58 MECHANICS. 

Measure of the tendency to Rotation about the Axes. 
46. Let X, Y, and Z denote the components of the 
resultant of the system, as in 
the last article, and denote, as £ 

before, the co-ordinates of the 3 ^ 

point of application of the re- 
sultant by #!, 2/t, and z v To find 
the resultant moment, with re- 
spect to the axis of Z, it may 

be observed that the component yy/i"_ y'j 

Z, can produce no rotary effect, ' x 

since it is parallel to the axis of Fig. 30. 

Z\ the moment of the compo- 
nent Y] with respect to the axis of Z, is Yx l ; the moment 
of the component X, with respect to the same axis, is 
—Xy^ the negative sign being taken because the force X 
tends to produce rotation in a negative direction. Hence, 
the resultant moment of the system, with respect to the 
axis of Z, is, 

Yx x - Xy x ; 

or, substituting for JTand Y their values, we have, 

Yx x — Xy x = 2(Pcos/3a* — Pcosay) . (24.) 

In like manner for the resultant moment of the system, 
with respect to the axis X, 

Zy l - Yz y — 2(Pcos/ y — Pcos/3 z) . (25.) 

And for the resultant moment, with respect to the axis 

of r; 

Xz l — Zx 1 = 2(Pcosas — jPcos/cc) . (26.) 

Equilibrium of Forces in a Plane. 

47. In order that a system of forces lying in the same 

plane, and applied at points of a free solid, may be in 

equilibrium, two conditions must be fulfilled: First, the 

lesultant of the system must have no tendency to produce 



EQUILIBRIUM OF FOECES. 59 

motion of translation; and, secondly, it must have nc 
tendency to produce motion of rotation. Conversely, if 
these conditions are satisfied, the system will be in equi- 
librium. 

The first condition will be fulfilled, and will only be ful- 
filled, when the resultant is equal to ; but from Art. 44, we 
have, 

The value of Jl can only be equal to when .X = 0, and 
Y = ; or, what is the same thing, 

2(Pcosa) = 0, and 2(Pcos/3) = . (27.) 

The second condition will be fulfilled, and will only be 
fulfilled, when the moment of the resultant, with respect to 
any point of the plane, is equal to 0, whence, 

Br = 0; or, 2(7» = . . . (28.; 

Hence, from Equations (2V) and (28), in order that a 
system of forces, lying in the same plane, and applied at 
points of a free solid body, may be in equilibrium, we must 
have, 

1st. The algebraic sum of the components of the forces in 
the direction of any two rectangular axes separately equal 
to 0. 

2d. The algebraic sum of the moments of the forces, with 
respect to any point in the plane, equal to 0. 

Equilibrium of Forces in Space. 
48. In order that a system of forces situated in any man- 
ner in space, and applied at points of a free solid body, may 
be in equilibrium, two conditions must be fulfilh d. First, the 
forces must have no tendency to produce motion of transla- 
tion ; and secondly, they must have no tendency to produce 
motion of rotation about either of the three rectangular 
axes. Conversely, when these conditions are fulfilled, the 
system will be in equilibrium. The first condition will be 



60 MECHANICS. 

fulfilled, and will only be fulfilled, when the resultant is 
equal to 0. But, from Equation (21), 



P = t/X 2 + Y* + Z\ 

That this value of P may be 0, we must have, separately, 

X = 0, Y = 0, and Z = ; 

or, what is the same thing, 

2(Pcosa) -0, 2(Pcos.3) = 0, and2(Pcos/) = . (29.) 

The second condition will be fulfilled, and will only be 
fulfilled, when the moments, with respect to each of the 
three axes, are separately equal to 0. This gives (Art. 46), 



2(Pcos/3a; — JPcomy) = 
2(Pcosyy — Pcos/3z) — f • 
2(Pcosas — JPcosyx) = Oj 



(30.) 



Hence (Equations 29 and 30), in order that a system of 
forces in space applied at points of a free solid may be in 
equilibrium : 

1st. The algebraic sum of the components of the forces in 
the direction of any three rectangular axes must be separate- 
ly equal to 0. 

2d. TJie algebraic sum of the moments of the forces, vnth 
respect to any three rectangular axes, must be separately 
equal to 0. 

Equilibrium of Forces applied to a Revolving Body. 
49. If a body is restrained by a fixed axis, about which 
it is free to revolve, we may take this line as the axis of X. 
Since the axis is fixed, there can be no motion of transla- 
tion, neither can there be any rotation about either of the 
other two axes of co-ordinates. All of Equations (29), and 
the first and third of Equations (30), will be satisfied b} 
virtue of the connection of the body with the fixed axil 



EQUILIBRIUM OF FORCES. 61 

The second of Equations (30) is, therefore, the only one that 
must be satisfied by the relation between the forces. We 
must have, therefore, 

l(Tcosyy - JPcoapz) = . . (31.) 

That is, if a body is restrained by a fixed axis, the forces 
applied to it will be in equilibrium when the algebraic sum 
of the moments of the forces with respect to this axis is 
equal to 0. 



(ft 



MECHANICS. 



CHAPTER III. 

CENTRE OF GRAVITY AND STABILITY. 

Weight. 

50. That force by virtue of which a body, when aban- 
doned to itself, falls towards the earth, is called the force. 
of gravity. The force of gravity acts upon every particle 
of a body, and, if resisted, gives rise to a pressure; this 
pressure is called the weight of the particle. The resultant 
weight of all the particles of a body is called the weight of 
the body. The weights of the particles are sensibly directed 
towards the centre of the earth; but this point being nearly 
4,000 miles from the surface, we may, for all practical pur- 
poses, regard these weights as parallel forces ; hence, the 
weight of a body acts in the same direction as the weights 
of its elementary particles, and is equal to their sum. 

Centre of Gravity. 

51. The centre of gravity of a body is the point of ap- 
plication of its weight. The weight being the resultant of 
a system of parallel forces, the centre of gravity is a centre 
of parallel forces, and so long as the relative position of the 
particles remains unchanged, this point will retain a fixed 
position in the body, and this independently of any parti- 
cular position of the body (Art. 43). The position of the 
centre of gravity is entirely independent of the value of the 
force of gravity, provided that we regard this force as con- 
stant throughout the dimensions of the body, which we may 
do in all practical cases. Hence, thfe centre of gravity is the 
same for the same body, wherever it may be situated. The 
determination of the centre of gravity is, then, reduced to 
the determination of the centre of a system of parallel 



CENTKE OF GRAVITT. 63 

forces. Equations (13) are, therefore, immediately appli- 

plicable. 

Preliminary discussion. 

52. Let there be any number of weights applied at 
points of a straight line. We may take the axis of X to 
coincide with this line, and because the points of application 
of the weights are on this line, we shall have, 

y = o, y' — 0, &c. ; 3. = 0, z' = 0, &c. ; 

substituting these in the second and third of Equations (13), 
we have, 

y x = 0, and Zj = 0. 

Hence, the point of application of the resultant is on the 
given line. 

In the case of a material straight line, that is, of a line 
made up of material points, the weight of each point will be 
applied at that point, and from what has just been shown, 
the point of application of the resultant weight will also be 
on the line ; but this point is the centre of gravity of the line. 

Hence, the centre of gravity of a material straight line 
is situated somewhere on the line. 

Let weights be applied at points of a given plane. We 
may take the plane JCY to coincide with this plane, and in 
this case we shall have, 

2 = 0, z' = 0, &c. ; 

these in the third of Equations (13) will give, 

z 1 = 0; 

hence, the point of application of the resultant iveights is 
in the plane. 

It may be shown, as before, that the centre of gravity of 
a material plane curve, or of a material plane area, is in 
the plane of the curve, or area. 

If the bodies considered are homogeneous in structure, 
the weights of any elementary portions are proportional to 



64 



MECHANICS. 



B 

Fig. 81. 



■SL 



B'A' 



their volumes, and the problem for finding the centre of 
gravity is reduced to that for finding the centre of figure. 
In what follows, lines and surfaces will be considered as made 
up of material points, and all the volumes considered will be 
regarded as homogeneous unless the contrary is stated. 

Centre of Gravity of a straight line. 

53. Let there be two material M 
points M and jV, equal in weight, 
and firmly connected by an inflexible 
line 3IJST. The resultant of these 
weights will bisect the line MN in S 
(Art. 40) ; hence S is the centre of 
gravity of the two points M and W. 

Let MJV be a material straight line, and JS its middle 
point. We may regard it as com- 
posed of heavy material points A, A' ; 
JB, B\ &c, equal in weight, and so 
disposed that for each point on one 
side of S, there is another point on 
the other side of it and equally distant 
from it. From what precedes, the 
centre of gravity of each pair of points is at $, and conse- 
quently the centre of gravity of the whole line is at IS. 
That is, the centre of gravity of a straight line is at its 
middle point. 

Centre of Gravity of symmetrical lines and areas. 

54. Let APBQ be a plane curve, and AB a diameter, 
that is, a line which bisects a system of 

parallel chords ; let JP Q be one of the 
chords bisected. The centre of grav- 
ity of the chord JP Q will be upon AB, 
and in like manner, the centre of 
gravity of any pair of points lying at 
the extremity of one of the parallel 
chords will be found upon the diam- 
eter ; hence, the centre of gravity of the entire curve is upon 
the diameter (Art. 52). The entire area of the curve is 



t 

Fig. 32. 




CENTRE OF GRAVITY. 65 

made up of the system of parallel chords bisected, and since 
the centre of gravity of each chord is upon the diameter, it 
follows that the centre of gravity of the area is upon the 
diameter. 

Hence, if any curve, or area, has a diameter, the centrf 
of gravity of the curve, or area, lies upon that diameter. 

If a curve or area has two diameters, the centre of gravitj 
will be found at their point of intersection. Hence, in the 
circle and ellipse the centre of gravity is at the centre of the 
curve. 

If a surface has a diametral plane, that is, a plane which 
bisects a system of parallel chords terminating in the surface, 
then will the centre of gravity of the extremities of each 
chord lie in the diametral plane, and consequently, the cen- 
tre of gravity of the surface will be in that plane. The 
centre of gravity of the volume bounded by such a surface, 
for like reason, lies in the diametral plane. 

Hence, if a surface, or volume, has a diametral plane, the 
centre of gravity of the surface, or volume, lies in that 
plane. If a surface, or volume, has three diametral planes 
intersecting each other in a point, that point is the centre 
of gravity. Hence, the centre of gravity of the sphere and 
the ellipsoid lie at their centres. We see, also, that the 
centre of gravity of a surface, or volume, of revolution lies 
in the axis of revolution. 

Centre of Gravity of a Triangle. 
55. Let ABC be any plane triang'e. Join the vertex 
A with the middle point D of the op- 
posite'side BC ; then will AD bisect A 

all of the lines drawn in the triangle /l\ 

parallel to the base BC; hence, the y./ / \ 

centre of gravity of the triangle lies / \ v f?. \ 

upon AD (Art. 54) ; for a like reason, c /.:Hv^£r : irgd^ 
the centre of gravity of the triangle D 

Fi 0, 34 

lies upon the line BE, drawn from 

the vertex B to the middle point of the opposite side AC ' 

it is, therefore, at G, their point of intersection. 



66 MECHANICS. 

Draw ED ; then, since ED bisects A G and B C, it is 
parallel to AD, and the triangles 
EGD and AGD are similar. The f 

side ED is equal to one-half of its // \ 

homologous side AD, consequently A^- ' \ 

the side GD is equal to one-half of /„Xi:- A 
its homologous side AG ; that is, the C ^~ : -"-"-"-"-"-^- -" -' r - A B 
point G is one-third of the distance . Figi 34. 
from D to A. 

Hence, the centre of gravity of a plane triangle is on a 
line drawn from the vertex to the middle point of the base, 
and at one-third of the distance from the base to the vertex. 

Centre of Gravity of a Parallelogram. 

56. Let A C be any parallelogram. Draw EF bisect- 
ing the sides AD and CD ; it will 

also bisect all lines of the parallelo- DE C 

gram parallel to these sides ; hence, the / '' / 

centre of gravity lies on it ; draw also / ~f " I s - 

the line OH bisecting the sides AD L 1 J 

and DC; for a similar reason, the Fi<r 35 

centre of gravity lies on it : it is, 
therefore, at G, their point of intersection. 

Hence, the centre of gravity of a parallelogram lies at 
the point of intersection of two straight lines joining the 
middle points of the opposite sides. 

It is to be remarked, that this point coincides with the 
point of intersection of the diagonals of the parallelogram. 

Centre of Gravity of a Trapezoid. 

57. Let AC he sl trapezoid. Join the middle points, 
and P, of the parallel sides, by a 

straight line ; this line will bisect all 
lines parallel to AD and D C ; hence, 
it must contain the centre of gravity. 
Draw the diagonal DD, dividing the 
trapezoid into two triangles. Draw 
also the lines DO and DP; take 




CENTRE OF GRAVITY. 



67 



OQ = \OD, and PR = \PB ; then will Q and R be 
the centres of gravity of these triangles (Art. 55). Join 
Q and J? by a straight line ; the centre of gravity of the 
trapezoid must be on this line (Art. 52). Hence, it is at 
G where the line QR cuts OP. 

Centre of Gravity of a Polygon. 

58. Let ABODE be any polygon, and a, 5, c, d, e, the 
middle points of its sides. The weights . 
of the sides will be proportional to 
their lengths, and may be represented 
by them. Let it first be required to 
find the centre of gravity of the peri- 
meter ; join a and 6, and find a point 
o, such that 

ao : ob : : BO : BA; 

then will o be the centre of gravity of the sides AB and 
BO. Join b and c, and find a point o\ such that 




oo' 



o'c :: CD : AB+BO; 



then will o' be the centre of gravity of the three sides, 
AB^ B (7, and OD. Join o' with d, and proceed as before, 
conthiuing the operation till the last point, 6r, is found ; this 
will be the centre of gravity of the perimeter. 

To find the centre of gravity of the area, divide it into 
the least number of triangles possible, and find the centre 
of gravity of each triangle. The weights of these triangles 
will be proportional to their areas, 
and may be represented by them. 
(Art. 52.) Let ABODE A be any 
polygon, and 0, 0\ 0", the centres of 
gravity of the triangles into which it 
can be divided. Join and 0\ and 
find a point 0'", such that 




O'O"' : 00'" :: ABC \ ACD; 



Fig. 38. 



88 



MECHANICS. 




then will 0'" be the centre of grav- 
ity of the two triangles ABC and 
A CD. 

Join 0" and 0"\ and find a point 
(*, such that 



0"'G : 0"G : : ABE : ^i?(7 4- ^4CZ>; 

then will G be the centre of gravity of the given polygon. 

Every curvilinear area may be regarded as polygonal, the 
number of sides being very great. Hence, the centres of 
gravity of their perimeters and areas may be found by the 
methods given. 

Centre of Gravity of a Pyramid. 

59. Any triangular pyramid may be regarded as made 
up of infinitely thin layers parallel to either of its faces. If 
a straight line be drawn from either vertex to the centre of 
gravity of the opposite face, it will pass through the centres 
of gravity of "all the layers parallel to that face. We may 
regard the weight of each layer as being applied at its cen- 
tre of gravity, that is, at a point of this line ; hence, the 
centre of gravity of the pyramid is on this line (Art. 52). 

Let AB CB be a pyramid, and K the middle point of 
BC. Draw KB and KA, and lay 
off KO = \KB, and KO' = \KA. 
Then will be the centre of gravity 
of the face BB C, and 0' that of the 
face GAB. Draw AO and BO' in- 
tersecting in G. Because the centre 
of gravity of the pyramid is upon both 
A and B 0\ it is at their intersection 
G. Draw 00'; then KO and KO' 
being respectively third parts of KB 
and KA, 0' is parallel to A B, and 
the triangles OGO and AGB are similar, consequently 




Fig. 



CENTRE OF GRAVITY. 69 

their homologous sides are proportional. But O r is one- 
third of AB, consequently OG is one-third of GA, or one- 
fourth of AO. 

Hence, the centre of gravity of a triangular pyramid is 
on a line drawn from its vertex to the centre of gravity of 
its base, and at one-fourth of the distance from the base to 
the vertex. 

Either face of a triangular pyramid may be taken as the 
base, the opposite vertex being considered as the vertex of 
the pyramid. 

To find the centre of gravity of a polygonal pyramid ; let 
A-BCDEF, represent any pyramid, A being the ver- 
tex. Conceive it divided into tri- 
angular pyramids, having a common -A 
vertex at A. If a plane be passed /h\ 
parallel to the base, and at one-fourth / /' \\ 
of the distance from the base to the / / ! \ \ 
vertex, it follows, from what has just BA — 7 / C~ "V\ti 
been shown, that the centres of gravi- \ // \\ / 

ty of all the partial pyramids will lie ^ ^/ 

in this plane. We may regard each Fig 40> 

pyramid as having its weight concen- 
trated at its centre of gravity ; hence, the centre of gravit y 
of the entire pyramid must lie in this plane (Art. 52). But 
it may be shown, as in the case of the triangular pyramid, 
that the centre of gravity lies somewhere in the line drawn 
from the vertex to the centre of gravity of the base ; it must, 
therefore, lie where this line pierces the auxiliary plane : 

Hence, the centre of gravity of any pyramid whatever 
lies on a line drawn from its vertex to the centre of gravity 
of its base, and ctt one-fourth of the distance from the base 
to the vertex. 

A cone is a pyramid having an infinite number of faces : 

Hence, the centre of gravity of a cone is on a line drawn 
from the vertex to the centre of gravity of the base, and at 
one-fourth of the distance from the base to the vertex. 



70 MECHANICS. 

Centre of Gravity of Prisms and Cylinders. 

60. Any prism whatever may be regarded as made up 
of layers parallel to the bases. If a straight line be drawn 
between the centres of gravity of the two bases, it will pass 
through the centres of gravity of all these layers. The 
centre of gravity of the prism will, therefore, lie somewhere 
in this line, which we may call the axis of the prism. We 
may also regard the prism as made up of material lines 
parallel to the lateral edges of the prism. If a plane be 
passed midway between the two bases and parallel to them, 
it will bisect all of these lines, and consequently their 
centres of gravity, as well as that of the entire prism, 
will lie in it. It must, therefore, be at the point in which 
the plane cuts the axis of the prism, that is, at its middle 
point. 

Hence, the centre of gravity of a prism is at the middle 
point of its axis. 

When the bases of the prism become polygons having an 
infinite number of sides, the prism will become a cylinder, 
and the principle just demonstrated will still hold good : 

Hence, the centre of gravity of a cylinder with parallel 
bases is at the middle point of its axis. 

Centre of Gravity of Polyhedrons. 

61. If any point within a polyhedron be assumed, and 
this point be joined with each vertex of the polyhedron, we 
shall thus form as many pyramids as the solid has faces : the 
centres of gravity of these pyramids may be found by the 
rules for such cases. If the centres of gravity of the first 
and second pyramid be joined by a straight line, the com- 
mon centre of gravity of the two may be found by a 
process entirely similar to that used in finding the centre of 
gravity of a polygon, observing that the weights of the par 
tial pyramids are proportional to their volumes, and that 
they may be represented by their volumes. Having com- 
pounded the weights of the first and second, and found it3 
point of application, we may, in like manner, compound this 



CENTRE OF GRAVITY. 71 

with the weight of the third, and so on, till the centre of 
gravity of the entire pyramid is determined. 

Any solid body bounded by a curved surface may be 
regarded as a polyhedron whose faces are extremely small, 
and its centre of gravity may be determined by the rule just 
explained. 

Experimental determination of the Centre of Gravity. 

63. We know that the weight of a body always passes 
through its centre of gravity, no matter what may be the 
position of the body. If we attach a flexible cord to a body 
at any point and suspend it freely, it must ultimately come 
to a state of rest. In this position, the body is acted upon 
by two forces : the weight, tending to draw the body towards 
the centre of the earth, and the tension of the cord, which 
resists this force. In order that the body may be in equili- 
brium, these forces must be equal and directly opposed. 
But the direction of the weight passes through the centre 
of gravity of the body ; hence, the tension of the string, 
which acts in the direction of the string, must also pass 
through the same point. This principle gives rise to the 
following method of finding the centre of gravity of a 
body. 

Let AB C represent a body of any form whatever. Attach 
a string to any point, C, of the body, 
and suspend it freely ; when the body 
comes to a state of rest, mark the di- 
rection of the string ; then suspend the 
body by a second point, B, as before, 
and when it comes to rest, mark the 
direction of the string ; their point of 
intersection, G, will be the centre of 
gravity of the body. 

Instead of suspending the body by a string, it may be 
balanced on a point. In this case, the weight acts vertically 
downwards, and is resisted by the reaction of the point ; 
hence, the centre of gravity must lie vertically over the 
point. 




72 MECHANICS. 

If, therefore, the body be balanced at any two points of 
its surface, and verticals be drawn through the point, in 
these positions, their intersection will be the centre of gravi- 
ty of the body. 

It follows, from what has just been explained, that when 
a body is suspended by an axis, it can only come to a state 
of rest when the centre of gravity lies in a vertical plane 
passed through the axis. 

The centre of gravity may lie above the axis, below the 
axis, or on the axis. 

In the first case, if the body be slightly deranged, it will 
continue to revolve till the centre of gravity falls below the 
axis ; in the second case, it will return to its primitive po- 
sition ; in the third case, it will remain in the position in 
which it is placed. These cases will be again referred to, 
under the head of Stability. 

The preceding rules enable us to find the centres of gravi- 
ty of all lines, surfaces, and solids ; but, on account of the 
difficulty of applying them in certain cases, we shall annex 
an outline of some of the methods, by the Differential and 
Integral Calculus. Those magnitudes whose centres of grav- 
ity are most readily found by the calculus, are mathematical 
curves ; areas bounded wholly, or in part, by these curves ; 
curved surfaces ; and volumes bound by curved surfaces. 

^Determination of the Centre of Gravity by means of the Calculus. 

64. To place Formulas (13) under a suitable form for the 
application of the calculus, we have simply to substitute for 
the forces P, P, &c, the elementary volumes, or the differ- 
entials of the magnitudes, and to replace the sign of summa- 
tion, 2, by that of integration, /. 

Making these changes, Formulas (13) become, 

_ fxdm _ fydm _ fzdm 

1 ~ fdm ' Vl ~ fdm ' Zl ~ /dm ' ^^ 

In which dm denotes the differential of the magnitude in 
question ; x, y, and 2, the co-ordinates of its centre of grav- 



CENTRE OF GRAVITY. 



73 



ity, and x x , y v and g 15 the co-ordinates of the centre of grav- 
ity of the magnitude. 

Application to plane curves. 
65. The plane XY may be taken to coincide with that 
of the curve, in which case, 2 = for every point of the 
curve ; and, consequently, z x = ; dm becomes the differ- 
ential of an arc of a plane curve, or dm = -y/dx? -\- dy" 1 . 
Substituting in (32), we have, 



fxydx 2 -}- dy" 1 
f -\/dx 2 + dy* 



Vi 



fy^/dx 1 ^ dy" 

f^d^TW 



(33.) 



Centre of Gravity of an arc of a circle. 
66. Let AB C be the arc, the origin of co-ordinates 
and centre of the circle, OX the axis of 
abscissas, perpendicular to the chord of 
the arc, and OY the axis of ordinates. 
Since the arc is symmetrically situated 
with respect to the axis of X, the centre 
of gravity is somewhere on this line 
(Art. 54) ; consequently, y x = 0. To find 
ajj, we have the equation of the circle, 




Differentiating, 



x 2 + 2/ 2 = r*. 



2xdx + 2ydy = 



<&> = ^df 



Substituting in the first of Formulas (33), and reducing, 
we find, 

frdy 



f 



rdy 



Vr*-y> 

Integrating both numerator and denominator between the 
limits y = — £c, and y = -f |c, we have, 



+ ic 



frdy —. 



re 



rr 


I 






MECHANICS. 




and, 














I 


rely 


= r sir 


<**- 


r sin * 


~2r 


H 


ence 


, by substitution, 














re 


or, arc 


ABC 






arc 


ABC 





arc ABC. 



That is, the centre of gravity of an arc of a circle is on 
the diameter which bisects its chord, and its distance from 
the centre is a fourth proportional to the arc, chord, and 
radius. 

Application to Plane areas. 

67. Let the plane JCY be taken to coincide with that 
of the area. We shall have, as before, z 1 — 0. In this 
case, we have elm = ydx ; and, consequently, Formulas 
(32), reduce to 



fydx 



fydx 



(34.) 



Centre of Gravity of a parabolic area. 

6§. Let A OB represent the area, 
having its chord at. right angles to the 
axis. Let be the origin of co-ordinates, 
taken at the vertex, and let the axis of J£ 
coincide with the axis of the curve ; the 
value of 2/j will, as before, be equal to 0. 
To find the value of x^, we have the equa- 
tion of the parabola, 




Fig. 43. 



y* = 2px .'. y — Jlp . x . 
By substitution in the: first of Formulas (34), we have, 

. 3 2. 

/V 2p.x 2 dx f x 2 elx 

Xl = ~ r- \ = ~T~' 
fy'2p .x 2 elx f x 2 elx 



CENTRE OF GRAVITY. 



7* 



Integrating between the limits x = 0, and x = a, we 
have, 



and, 



hence, 



J x 3 dx = |a , 



/ a 2 d# 
o 



= *J 



«i = I «• 



That is, *Ae c^n^re o/ gravity of a segment of a parabola 
is on its axis, and at a distance from the vertex equal to 
three-fifths of the altitude of the segment. 



Application to solids of revolution. 

69. If we take the planes XY and XZ passing through 
the axis of revolution, the centre of gravity will lie in both 
these planes, therefore y l and z l will both be 0. In this 
case, the first of Formulas (32) will be sufficient. 

Since the axis of X coincides with the axis of revolution, 
dm becomes equal to irifdx. Substituting in the first of 
Formulas (32), we have, 



/ xy*dx 
fy'dx 



(35.) 



Centre of Gravity of a semi-ellipsoid. 

70. Let the semi-ellipse A GB, be re- 
volved about the axis 0(7; it will gener- 
ate a semi -ellipsoid whose axis coincides 
with the axis of X. Both y x and z 1 
being 0, it only remains to find the value 
of x v 




Fig. 44. 



76 MECHANICS. 

The equation of the ellipse referred to its centre, is, 

m which a and b are the semi-axes. 
Substituting, in Equation (35), we have, 

C —(a 2 x — x 3 )dx f(a?x — x*)dx 

«i = — | = 

f —(a 2 — x*)dx f (a* — x*)dx 

Integrating between the limits, x = 0, and x = a, we 
have 



and, 

a 



Substituting, we have, 

a 4 2a 3 3 3 

That is, the centre of gravity of a semi-prolate spheroid 
of revolution is on its axis of revolution, and at a distance 
from the centre equal to three-sixteenths of the major axis 
of the generating ellipse. 

The examples above given are enough to indicate the 
method of applying the calculus to the determination of the 
centre of gravity. 

Centre of Gravity of a system of bodies. 
71. When we have several bodies, and it is required to 
find their common centre of gravity, it will, in general, be 
found most convenient to employ the principle of moments. 



CENTRE OF GRAVITY. 77 

To do this, we first find the centre of gravity of each body 
separately, by the rules already given. The weight of each 
body may then be regarded as a force applied at the centre 
of gravity of the body. The weights being parallel, we 
have a system of parallel forces, whose points of application 
are known. If these points are all in the same plane, we 
may find the lever arms of the resultant of all the weights, 
with respect to two lines, at right angles to each other in 
that plane ; and these will make known the point of applica- 
tion of the resultant, or, what is the same thing, the centre 
of gravity of the system. If the points are not in the same 
plane, the lever arms of the resultant of all the weights may 
be found, with respect to three axes, at right angles to each 
other ; these will make known the point of application of the 
resultant weight, or the required position of the centre of 
gravity. 

MISCELLANEOUS EXAMPLES. 

1. Required the point of application of the resultant of 
three equal weights, applied at the three vertices of a plane 
triangle. 

SOLUTION. 

Let ABC (Fig. 34) represent the triangle. The resul- 
tant of the weights applied at B and C will be applied at 
Z>, the middle point of B C. The weight acting at JD being 
double that at A, the total resultant will be applied at G, 
making GA = 2 GD ; hence, the required point is at the 
centre of gravity of the triangle. 

2. Required the point of application of the resultant of 
a system of equal parallel forces, applied at the vertices of 
any regular polygon ? 

Ans. At the centre of gravity of the polygon. 

3. Parallel forces of 3, 4, 5, and 6 lbs., are applied at the 
successive vertices of a square, whose side is 12 inches. At 
what distance from the first vertex is the point of applica- 
tion of their resultant ? 



78 MECHANICS. 

SOLUTION. 

Take the sides of the square through the first vertex as 
axes of moments ; call the side through the first and second 
vertex the axis of -5T, and that through the first and fourth 
the axis of Y. We shall have from Formulas (13), 

4X12 + 5X12 

* = f 8 = 6; 

-. 6X12 + 5X12 22 

and x, = ' = 

1 18 3 

Denoting the required distance by d, we have, 



d = vV + 2/i 2 = 9.475 in. Ans. 

4. Seven equal forces are applied at seven of the vertices 
of a cube. What is the distance of the point of application 
of their resultant from the eighth vertex ? 

SOLUTION. 

Take the eighth vertex as the origin of co-ordinates, and 
the three edges passing through it as axes of moments. We 
shall have from Equations (13), denoting one edge of the 

cube by #, 

x 1 = ± a , y Y — * a , and z x == fa. 

Denoting the required distance by d, we have, 

d = v^i 2 + Vi + z \ = * a V%- Ans ' 

5. Two isosceles triangles are constructed on opposite 
sides of the base b, having altitudes respectively equal to 
h and A', h being greater than h'. Where is the centre of 
gravity of the space lying within the two triangles ? 

SOLUTION. 

It must lie on the altitude of the greater triangle. Take 
the common base as an axis of moments; then will the 
moments of the triangles be, respectiv V, \bh x |A, and 



CENTRE OF GRAVITY. 79 

\bh r x \h f ; and from the first of Formulas (13), we shall 
have, 

(B i- £J(A + A') - >( k ~ h >- 

That is, the required centre of gravity is on the altitude 
of the greater triangle, at a distance from the common base 
equal to one-third of the difference of the two altitudes. 

6. When is the centre of gravity of the space included 
between two circles tangent to each other internally ? 

SOLUTION. 

Take their common tangent as an axis of moments. The 
centre of gravity will lie on the common normal, and its 
distance from the point of contact is given by the equation, 

at 9 — tr' % r* + rir' + ?*' 2 

x, — 



irr* — <sr' a r -\- r' 

7. Let there be a square, and suppose it divided by its 
diagonals into four equal parts, one of which is removed. 
Required the distance of the centre of gravity of the re- 
maining figure from the opposite side of the square. 

Ans. T 7 ¥ of the side of the square. 

8. To construct a triangle, having given its base and 
centre of gravity. 

SOLUTION. 

Draw through the middle of the base, and the centre of 
gravity, a straight line ; lay off beyond the centre of gra- 
vity a distance equal to twice the distance from the middle 
of the base to the centre of gravity. The point thus found 
is the vertex. 

9. Given the base and altitude of a triangle. Required 
the triangle, when its centre of gravity is perpendicularly 
over the extremity of the base. 

10. Three men carry a cylindrical bar, one taking hold 




80 MECHANICS.* 

of one end, and the others at a common point. Required 
the position of this point, in order that the three may sus- 
tain equal portions of the weight. 

Pressure of one body upon another. 

72. Let i k a movable body 
pressed against a fixed body B, 
and touching it at a single point. 
In order that A may be in equi- 
librium, the resultant of all the 
forces acting upon it, including its 
weight, must pass through the point 
of contact, P' ; otherwise there would 
be a tendency to rotation about JP\ Fi 45 
which would be measured by the 

moment of the resultant with respect to this point. Fur- 
thermore, the direction of the resultant must be normal to 
the surface of B at the point P', else the body A would 
have a tendency to slide along the body J5, which tendency 
would be measured by the tangential component. The 
pressure upon B develops a latent force of reaction, which 
must be equal and directly opposed to it. The resultant of 
all the forces must be equal to zero (Art. 4*7). That is, 
when a body, resting upon another and acted upon by any 
number of forces is in equilibrium, the resultant of all the 
forces called into play is equal to 0. 

If all the forces called into play are taken into account, 
the algebraic sums of their moments with respect to any 
three rectangular axes will be separately equal to 0. 

Equations (29) and (30) are, then, perfectly general in 
every case of equilibrium, provided all of the forces called 
into play are taken into account. 

Stable, Unstable, and Indifferent Equilibrium. 

73. A body is in stable equilibrium when, on being 
slightly disturbed from its state of rest it has a tendency to 



STABILTT. 



81 



% 



Fig. 46. 



return to that state. This will, in general, be the case when 

the centre of gravity of the body is at its lowest point. Let 

A be a spherical body suspended from 

an axis 0, about which it is free to 

turn. When the centre of gravity of 

A lies vertically below the axis, it is 

in equilibrium, for the weight of the 

body is exactly counterbalanced by the 

resistance of the axis. Moreover, the 

equilibrium is stable; for if the body 

be deflected to A\ its weight tends to 

restore it to its position of rest, A. The measure of this 

tendency is W x OP, that is, the moment of the weight with 

respect to the axis 0. Under the action of the force W, the 

body will return to A, and, passing to the other side by 

virtue of its inertia, will finally come to rest and return 

again to A', and so on, till after a few vibrations, when it 

will come to rest at A. 

A body is in unstable equilibrium Avhen, being slightly 
disturbed from its state of rest, it tends to depart still far- 
ther from it. This will, in general, be the case when the 
centre of gravity of the body occupies its highest position. 

Let A be a sphere, connected by an 
inflexible rod with the axis 0. When 
the centre of gravity of A lies verti- 
cally above 0, it will be in unstable 
equilibrium ; for, if the sphere be de- 
flected to the position A', its weight 
will act with the lever arm OP to in- 
crease this deflection. The motion will 
continue till, after a few vibrations, it comes to rest 
below the axis. In this last position, it will be in stable 
equilibrium. 

A body is in indifferent, or neutral, equilibrium when it 
remains at rest wherever it may be placed. This will, in 
general, be the case when the centre of gravity continues in 
the same horizontal plane on being slightly disturbed. 



rr 



o# 



__Jp 

Fig. 47. 



82 



MECHANICS. 




Let A be a sphere, supported 
by a horizontal axis OP passing 
through its centre of gravity. 
Then, in whatever position it may 
be placed, it will have no tendency 
to change this position ; it is, therefore, in indifferent, 01 
neutral equilibrium. 

In the figure, A, B, and C represent a cone in positions 
of stable, unstable, and indifferent equilibrium. 



Fig. 48. 




Fig. 49. 

If a wheel, or other solid, be mounted on a horizontal axis, 
about which it is free to turn, the centre of gravity not lying 
on the axis, it will be in stable equilibrium, when the centre of 
gravity is directly below the axis ; and in unstable equi- 
librium when it is directly above the axis. When the axis 
passes through the centre of gravity, it will, in every po- 
sition, be in neutral equilibrium. 

We infer, then, from the preceding discussion, that when 
a body at rest is so situated that it cannot be disturbed from 
its position without raising its centre of gravity, it is in a 
state of stable equilibrium / when a slight disturbance de- 
presses the centre of gravity, it is in a state of unstable equi- 
librium; when the centre of gravity remains constantly in the 
same horizontal plane, it is in a state of neutral equilibrium. 

This principle holds true in combinations of wheels, as in 
machinery, and indicates the importance of balancing the 
elements, so that their centres of gravity may remain aa 
nearly as possible in the same horizontal planes. 



STABILITY. 83 

Stability of Bodies on Horizontal Planes. 

74. A body resting on a horizontal plane may touch it 
in one, or in more than one point. k 
In the latter case, the salient poly- /! \\ 
gon, formed by joining the extreme / — 7 J A \t-t — y 
points of contact, as abed, is called / j/_\/ / 
the polygon of support. L S 

Fig. 50. 

When the direction of the weight of the body, that is, the 
vertical through its centre of gravity, pierces the plane within 
the polygon of support, the body is stable, and will remain in 
equilibrium, unless acted upon by some other force than the 
weight of the body. In this case, the body will be most 
easily overturned about that side of the polygon of support 
which is nearest to the line of direction of the weight. The 
moment of the weight, with respect to this side, is called 
the moment of stability of the body. Denoting the weight 
of the body by W, the distance from the line of direction 
of the weight to the nearest side of the polygon of support, 
by r, and the moment of stability by JS, we have, 

jS = Wr. 

The moment of stability is equal to the least moment of 
any extraneous force which is capable of overturning the 
body in any direction. The. weight of the body remaining 
the same, its stability will increase with r. If the polygon 
of support is a regular polygon, the stability will be great- 
est, other things being equal, when the direction of the 
weight passes through its centre. The area of the polygon 
of support remaining constant, the stability will be greater 
as the polygon approaches a circle. The polygon of support 
being regular, but variable in area, the stability will increase 
as this area increases. Hence, low bodies resting on ex- 
tended bases, are, other things being equal, more stable than 
high bodies resting on narrow bases. 

When the direction of the weight passes without the 
polygon of support, the body is unstable, and unless sup 






84 MECHANICS. 

ported by some other force than the weight, it will overturn 
about that side which is nearest to the direction of the 
weight. In this case, the product of the weight into the 
shortest distance from its direction to any side of the poly- 
gon, is called the moment of instability. Denoting this 
moment by 7", we have, as before, 

I=z Wr. 

The moment of instability is equal to the least moment 
of any force which can be applied to prevent the body from 
overturning. 

If the direction of the weight intersect any side of the 
polygon of support, the body will be in a state of equili- 
brium bordering on rotation about that side. 

The stability of a body will be greater, the more nearly 
the resultant of all the forces acting upon it, including its 
weight, is to being normal to the bearing surface. A 
maximum stability will be obtained, other things being 
equal, when the resultant is exactly perpendicular to the 
bearing surface. These principles find application in most 
of the arts, but more especially in Engineering and Architec- 
ture. In structures of all kinds intended to be stable, the 
foundation should be as broad as is consistent with the gen- 
eral design of the work, that the polygon of support may 
be as great as possible. The pieces for transmitting pres- 
sures should be so combined that the pressures transmitted 
to the ultimate polygons of support should be as nearly 
normal to the bearing surfaces as possible, and their lines 
of direction should pass as near the centres of the polygons 
of support as may be. The same principles hold good at 
all the points of junction between pieces employed for 
transmitting pressures. Hence, joints should be made as 
nearly normal to the pressures as possible. 

In the construction of machinery the preceding principles 
also apply. The centres of gravity of the rotating pieces 
should be on their axes, otherwise there will result an irre- 
gularity of motion, which, besides making the machine 



STABILITY. 



85 



work imperfectly, will ultimately destroy the parts of the 
machine itself. 

In loading cars, wagons, &c, we should endeavor to 
throw the centre of gravity of the load as near the track 
as possible. This is, in practice, partially effected by placing 
the heavier articles at the bottom of the load. 

It is needless to enumerate the multitudinous applications 
of the principles of stability ; they are of continual occur- 
rence in the daily transactions of life. 



PRACTICAL PROBLEMS IN CONSTRUCTION. 



1. A horizontal beam AB, 
which sustains a load, is sup- 
ported upon a pivot at A, and 
by a cord BE, the point E 
being vertically over A. Re- 
quired the tension of the cord 
BE, and the vertical pressure 
on the pivot A. 

SOLUTION. 

Denote the weight of the beam, together with its load, 
by W, and suppose its point of application to be at C. 
Denote C A hj p, and the perpendicular distance AF, from 
A to DE, hyp'. Denote also the tension of the cord by t. 
If we regard A as the centre of moments, we shall have, in 
the case of an equilibrium, 




Wp = tp'\ 



= W^ 



Or, denoting the angles EDA by a, and the distance AB 
by b, we shall have, 

P 



W, 



6sina 



To find the vertical pressure on the pivot A, resolve the 
force t into two components, respectively parallel and per 



86 MECHANICS. 

pendicular to A B. We shall have for the latter component 
denoted by t\ 

t' = t sina = W^- • 
o 

The vertical pressure upon A, plus the weight TFJ must 
be equal to this value of t'. Denoting this pressure by P, 
we shall have, 

When D C = ; or, when D and G coincide, the vertical 
pressure becomes 0. 

2. A rope AD, supports a pole, DO, of uniform thick- 
ness, one end of which rests upon a 
horizontal plane, and from the other D 

end is suspended a weight W. Re- y$ 

quired the tension of the rope, and y\//}-f- 

the thrust, or pressure, on the pole, j^s^ // ^ v 

the weight of the pole being neg- ■ " 

lected. Fig 5 2. 

solution. 

Denote the tension of the rope by t, the pressure on the 
pole by p, the angle ADO by a, and the angle ODW 
by/3. 

There are three forces acting at D, which hold each other 
in equilibrium ; the weight W, acting downwards, the ten- 
sion of the rope acting from D, towards A, and the thrust 
of the pole acting from toAvards D. Lay off Del, to 
represent the weight, and complete the parallelogram of 
forces doaD ; then will Da represent the tension of the 
rope, and Do the thrust on the pole. 

From Art. 35, we have, 

_. . n x . _„_. sin /3 

t W : : sin p : sin a ; \ t = W —— — 

sin a 



STABILITY. 

We have, also, from the same principle, 
p : W : : sin(a + /3) ; sin a ; /. p = W 



87 



sin(a -f- fi) 
sin a. 



if the rope is horizontal, we shall have a == 90° — /3, 
which gives, 

W 

t = W tan/3 , and p — 5 • 

3. A beam AB, is suspended by two ropes attached at 
jts extremities, and fastened to pins A and H. Required 
the tensions upon the ropes. 




SOLUTION. 

Denote the weight of the beam 
and its load by W, and suppose that 
G is the point of application of this 
force. Denote the tension of the 
rope BH, by t, and that of the rope 
FA, by I'. The forces acting to produce an equilibrium, 
are W, t, and t ' . The plane of these forces must be verti- 
cal, and further, the directions of the forces must intersect 
in a point. Produce AF, and BIT, till they intersect in K, 
and draw KG\ lay off KG, to represent the weight of the 
beam and its load, and complete the parallelogram of forces, 
Kb Cf ; then will Kb represent t, and Kf will represent t'. 
Denote the angle CKB by a, and the angle GKF by /3. 
We shall have, as in the last problem, 



W: t 
And, 
W: t' 



sin(a + (3) : sin/3 ; 



sin (a -f- (3) : sin a ; 



sinp 



*'= w 



sin (a + (3) 



sma 



4 A gate All, is supported at upon a pivot, and at 
A by a hinge, attached to a post AB. Required the 
pressure on the pivot, and also the tension of the hinge. 



88 



MECHANICS. 



/ I* 
|/ i . 

O/ — |c 



SOLUTION. 

Denote the weight of the gate and 
its load, by W. Produce the vertical 
through the point of application (7, of 
the force W, till it intersects the hori- 
zontal through A in D, and draw the 
line DO. Then will DA and DO 
represent the directions of the requir- 
ed components of W. Lay off Dc, 
to represent the value of W, and 
complete the parallelogram of forces, Dcoa ; then will Do 
represent the pressure on the pivot 0, and Da the pres- 
sure on the hinge, A. Denoting the angle oDc by &, the 
pressure on the pivot byjo, and on the hinge by p\ we shall 
have, 

W p 



o :e 

Fig. 51 



P = 



and p' = 

cosa ' M smx 



If we denote the distance OE by 5, and the distance DE 

by h, we shall have, 

h ' . b 

cos a =z — , ana sin a = — — • 

Hence, 

TTi/6 2 + A 2 , , p #+1" 

^ = y —- -, andy = - JLJL -i; 



h 



5. Having given the two 
rafters AC and DC of a 
roof, abutting in notches of 

a tie-beam AB, it is required ^r 

to find the pressure, or thrust, 
upon the rafters, and the di- 
rection and intensity of the 
pressure upon the joints at the tie-beam. 




Fig. 55. 



SOLUTION. 



Denote the weight of the rafters and their load by 2w ; 
we may regard this weight as made up of three parts— a 



STABILITY. 89 

weight w, applied at (7, and two equal weights \w, applied 
at A and B respectively. Let us denote the half span AB 
by s, the rise CB by A, and the length of the rafter A C or 
CB by I. Denote, also, the pitch of the roof CBB by a, 
the thrust on the rafter by t, and the resultant pressure at 
each of the joints A and B by p. 

Lay off Co to represent the weight to, and complete the 
parallelogram of forces Cboa ; then will Ca and Cb repre- 
sent the thrust upon the rafters ; and, since the figure Cboa 
is a rhombus, we shall have, 

w wl 

t sina = ±w .'. t = -——. — = — • 
J 2 sina 2A 

Conceive the force t to be applied at A, and resolve it into 
two components respectively parallel to CB and BA ; we 
shall have for these components, 

1 a * ws 
t sina = -k£, ana t cosa = — - • 

2 ' 2A 

The latter component gives the strain on the tie-beam, 
AB. 

To find the pressure on the joint, we have, acting down- 
wards, the forces \w and \w, or the single force w, and, act- 
ing from B towards A, the force — j- ; hence, 



If we denote the angle BAB by /3, we shall have from 
the right-angled triangle BAB, 

^ p BB s 

tan p = —r— = — - . 
AB 2h 

The direction of the joint should be perpendicular to that 
of the force p, that is, it should make with the horizon an 

angle whose tangent equals — • 

Lib 



90 MECHANICS. 

6. In the last problem suppose the rafters to aout against 
the wall. Required the least thickness that must be given 
to the wall to prevent it from being overturned. 

SOLUTION. 

Denote the entire weight thrown upon the wall by w, the 
length of that portion of the wall which sustains the pressure 
p by l\ its height by h\ its thickness by x, and the weight 
of each cubic foot of the material of the wall by w' ; then 
will the weight of this part of the wall be equal to w'h'l'x. 

The force — acts with an arm of lever h' to overturn the 

wall about its lower and outer edge ; this force is resisted by 
the weight w + w'h'l'x, acting through the centre of gravity 
of the wall with a lever arm equal to \x. If there is an 
equilibrium, the moments of these two forces must be equal, 

that is, — 7 x h — \w + w hi x) -, or = wx + w hi ar • 

Zth 2i lb 

Reducing, we have, x* -\ rnr.a; = . T ., ; 

to ' w hi' w I h 



P 

< «fS« 



w I WS w* 

° r ' x= ~ 2w'h'l' ± V ^M + «T ' 

7. A sustaining wall has a cross section in the form of a 
trapezoid, the face upon which the 

pressure is thrown being vertical, and A B 

the opposite face having a slope of 
six perpendicular to one horizontal. 
Required the least thickness that must 
be given to the wall at the top, that — = 
it may not be overturned by a nori- Fi 5 „ 

zontal pressure, whose point of appli- 
cation is at a distance from the bottom of the wall equal to 
one-third of its height. 

SOIUTION. 

Pass a plane through the edge A parallel to the face 
JB (7, aud consider a portion of the wall whose length is one 



STABILITY. 



91 



foot. Denote the pressure upon this portion by P, the 
height of the wall by 6A, its thickness at the top by x, and 
the weight of a cubic foot of the material by w. Let fall 
from the centres of gravity and 0' of the two portions, 
the perpendiculars OG and 0' E, and take the edge D as 
an axis of moments. The weight of the portion ABCF is 
equal to 6whx, and its lever arm, DG, is equal to h + \x. 
The weight of the portion AJDFis Sioh s , and its lever arm, 
DF, is f h. In case of an equilibrium, the sum of the mo- 
ments of their weights must be equal to the moment of P, 
whose lever arm is 2h. Hence, 

Qwhx(h + Ijb) + 3wh* X f h = P X 2h ; 

Qwhx + 3wx 2 + 2wk i = 2P. 



or. 



Whence, 



f + rt,.,^^, 



£B - 



V 3^ 



fA s 




8. Required the conditions of stability of a 
square pillar acted upon by a force oblique 
to the axis of the pillar, and applied at the 
centre of gravity of the pillar's upper 
base. 

SOLUTION. 

Denote the intensity of the oblique force 
by P, its inclination to the vertical by a, HI 1 a 

the length or breadth of the pillar by 2-a, Fi g- 57 - 

its height by x, and the weight of the pillar by W. Through 
the centre of gravity of the pillar draw the vertical A C, 
and lay off A C equal to W\ prolong PA and lay off AB equal 
to P ; complete the parallelogram of forces ABD C, and 
prolong the diagonal till it intersects HG or HG produced. 
If the point F falls between H and 6r, the pillar will be 
stable ; if it falls at H, it will be indifferent ; if it fills with- 
out IT, it will be unstable. To find an expression for tho 



92 MECHANICS. 

distance EG, draw DE perpendicular to A G. From the 
similar triangles APE and AEG, we have, 

AE: AG : : DE : EG; /. FG = ^^^-; 

AE 

But .4# = #, P.E' = Psina, and AE — W + Pcosa, 

hence we have, 

Px sina 



P 7 *? 



W + Pcosa' 



And, since .ZZTx equals a, we have the following condi- 
tions for stability, indifference and instability, respectively, 

Px sina 
a > 



a = 



a< 



W+ Pcosa' 

Pa? sina 
W + Pcosa ' 

P& sina 

TF-f- Pcosa' 



If we denote the distance EG by y, and the weight of a 
cubic foot of the material of the pillar by W, we shall have, 
since W = itfxw, 

Psina x 

u Aitfwx + Pcosa 

If, now, we suppose the intensity and direction of the 
force P to remain the same, whilst x is made to assume 
every possible value from up to any assumed limit, the 
value of y will undergo corresponding changes. The suc- 
cessive points thus determined make up a line which is 
called the line of resistance, and whose equation is that just 
deduced. 

If the pillar is made up of uncemented blocks, it will re- 
main in equilibrium so long as each joint is pierced by the 
line of resistance, provided that the tangent to the line of 
resistance makes with the normal to the joint an angle less 
than the limiting angle of resistance (Art. 103). 



STABILITY. 93 

The highest degree of stability will be attained when the 
line of resistance is normal to every joint, and when it 
passes through the centre of gravity of each. 

9. To determine the conditions of equilibrium and sta- 
bility of an arch of uncemented stones. 

SOLUTION. 

Let MNLK represent half of an arch sustained in equi- 
librium by a horizontal force JP, 
and by the weight of the arch- -- — ,~HL 

stones. Through the centre of / S^}~A^ m 

gravity of the first arch-stone draw /L ,A/ 

a vertical line, and on it lay off a /^Lf 

distance to represent the weight n( I 

of that stone. Prolong the direc- K~T^I 
tion of P, and lay off a distance rig> 58 - 

equal to the horizontal pressure. 

Complete the parallelogram of forces, aobB, and draw the 
diagonal oJB. This will be the resultant of the forces com- 
bined. Combine this resultant with the weight of the 
second arch-stone, and this with the weight of the third, 
and so on, till the last inclusive. The polygon oBCDE, 
thus found, is the line of resistance, and if this lies wholly 
within the solid part of the arch, the arch will be stable ; 
but, if it does not lie within it, the arch will be unstable. 
A rupture will take place at the joint where the line of re- 
sistance passes without the solid part of the arch. 

This problem may be solved analytically, in accordance 
with the principles already illustrated. It is only intended 
to indicate the general method of proceeding. 



94 



MECHANICS. 



CHAPTEK IY. 

ELEMENTARY MACHINES. 

Definitions and General Principles. 

75. A machine is a contrivance by means of which a 
force applied at one point is made to produce an effect at 
some other point. 

The force applied is called the power, and the point at 
which it is applied, is called the point of application. The 
force to be overcome is called the resistance, and the point 
at which it is to be overcome is called the working point. 

The working of any machine requires a continued applica 
tion of power. The source of this power is called the motor \ 

Motors are exceedingly various. Some of the most im- 
portant are muscular effort, as exhibited by man and beast 
in various kinds of work ; the weight and living force of 
water, as exhibited in the various kinds of water-mills ; the 
expansive force of vapors and gases, as displayed in steam 
and caloric engines ; the force of air in motion, as exhi- 
bited in the windmill, and in the propulsion of sailing 
vessels ; the force of magnetic attraction and repulsion, as 
shown in the magnetic telegraph and various magnetic 
machines ; the elastic force of springs, as shown in watches 
and various other machines. Of these motors, the most 
important ones are steam, air, and water power. 

To work, is to exert a certain pressure through a certain 
distance. The measure of the quantity of work performed 
by any force, is the product obtained by multiplying the 
effective pressure exerted, by the distance through which it 
is exerted. 

Machines serve simply to transmit and modify the action 
of forces. They add nothing to the work of the motor; on 



ELEMENTARY MACHINES. 95 

the contrary, they absorb and render inefficient much of the 
work that is impressed upon them. For example, in the 
case of a water-mill, only a small portion of the toork ex- 
pended by the motor is transmitted to the machine, on 
account of the imperfect manner of applying it, and of this 
portion a very large fraction is absorbed and rendered prac- 
tically useless by the various resistances, so that, in reality, 
only a small fractional portion of the work expended by the 
motor becomes effective. 

Of the applied work, a part is expended in overcoming 
friction, stiffness of cords, ba?ids, or chains, resistance of 
the air, adhesion of the parts, &c. This goes to wear out 
the machine. A second portion is expended in overcoming 
sudden impulses, or shocks, arising from the nature of the 
work to be accomplished, as well as from the imperfect con- 
nection of the parts, and from the want of hardness and 
elasticity in the connecting pieces. This also goes to strain 
and wear out the machine, and also to increase the sources 
of waste already mentioned. There is often a waste of 
work arising from a greater supply of motive power than is 
required to attain the desired result. Thus, in the move- 
ment of a train of cars on a railroad, the excess of the work 
of the steam, above what is just necessary to bring the train 
to the station, is wasted, and has to be consumed by the 
application of brakes, an operation which not only wears out 
the brakes, but also, by creating shocks, injures and ulti- 
mately destroys the cars themselves. 

Such are some of the sources of the loss of work. A 
part of these may, by judicious combinations and appliances, 
be greatly diminished ; but, under the most favorable cir- 
cumstances, there must be a continued loss of work, which 
requires a continued supply of power from the motor. 

In any machine, the quotient obtained by dividing the 
quantity of useful, or effective work, by the quantity of 
applied work, is called the modulus of the machine. As the 
resistances are diminished, the modulus increases, and the 
machine becomes more perfect. Could the modulus ever 



96 



MECHANICS. 



become equal to 1, the machine would be absolutely perfect 
Once set in motion, it would continue to move forever, 
realizing the solution of the problem of perpetual motion. 
It is needless to state that, until the laws of nature are 
changed, no such realization need be looked for. 

In studying the principles of machines, we proceed by 
approximation. For a first result, it is usual to neglect the 
effect of hurtful resistances, such as friction, adhesion, stiff- 
ness of cords, &g. Having found the relations between the 
power and resistance under this hypothesis, these relations 
are afterwards modified, so as take into account the various 
resistances. We shall, therefore, in the first instance, regard 
cords as destitute of weight and thickness, perfectly flexible, 
and inextensible. We shall also regard bars and connecting 
pieces as destitute of weight and inertia, and perfectly rigid ; 
that is, incapable of compression or extension by the forces 
to which they may be subjected. 

Elementary Machines. 

76. The elementary machines are seven in number — 
viz., the cord / the lever / the inclined plane y the pidley, a 
combination of the cord and lever; the wheel and axle, also a 
combination of the cord and lever ; the screw, a combination 
of two inclined planes twisted about an axis ; and the wedge, 
a simple combination of two inclined planes. It may easily 
be seen that there are in reality but three elementary 
machines — the cord, the lever, and the inclined plane. It 
is, however, more convenient to consider the seven above- 
named as elementary. By a suitable combination of these 
seven elements, the most complicated pieces of mechanism 

are produced. 

The Cord. 

77. Let AB represent a cord solicited by two forces 
P and M, applied at its extremi- 
ties, A and B. In order that p ^ \ % ^ R 

the cord may be in equilibrium, Fi 59 

it is evident, in the first place, 

that two forces must act in the direction of the cord, and in 



ELEMENTARY MACHINES. 97 

such a manner as to stretch it, otherwise the cord would 
bend under the action of the forces. In the second place, 
the intensitie's of the forces must be equal, otherwise the 
greater force would prevail, and motion would ensue. 
Hence, in order that two forces applied at the extremities 
of a cord may be in equilibrium, the forces must be equal 
and directly opposed. 

The measure of the tension of the cord, or the force by 
which any two of its adjacent particles are urged to sepa- 
rate, is the intensity of one of the equal forces, for it is 
evident that the middle point of the cord might be fixed and 
either force withdrawn, without diminishing or increasing 
the tension. When a cord is solicited in opposite directions 
by unequal forces directed along the cord, the tension will 
be measured by the intensity of the lesser force. 

Let AB represent a cord solicited by two groups of forces 
applied at its two extrem- 
ities. In order that these \ 

forces may be in equilibrium, '*"/ 

the resultant of the group ap- ^ 

plied at A and the resultant of 

the group at B must be equal and directly opposed. Hence, 
if we suppose all of the forces at each point to be resolved into 
components respectively coinciding with, and at right angles 
to AB, the normal components at each of the points must 
be such as to maintain each other in equilibrium, and the 
resultants of the remaining components at each of the points 
A and B must be equal and directly opposed. 

Let ABGD represent a cord, at the different points 
A, B, C, D, of which are 
applied groups of forces. If 
these forces are in equili- 
brium through the interven- 
tion of the cord, there must 
necessarily be an equili- 
brium at each point of ap- 
plication. Denote the tension of AB, BC, CD, by t, t', t'\ 
5 




98 MECHANICS. 

and the forces applied by JP, P\ P'\ &c, as shown m th«j 
figure. The forces in equilibrium about the point A are 
P, P\ P", and t, directed from A to B ; the forces in equili- 
brium about B are P" f , P lv , t, directed from B to A, and 
£', directed from B to (7. The tension t is the same at all 
points of the branch AB, and, since it acts at A in the direc- 
tion AB, and at B in the direction BA, it follows that 
these two forces exactly counterbalance each other. If, 
therefore, the forces P, P', P", were transferred from A to 
J5, unchanged in direction and intensity, the equilibrium at 
that point would be undisturbed. In like manner, it may 
be shown that, if all the forces now applied at B be trans- 
ferred to (7, without change of direction or intensity, the 
equilibrium at C would be undisturbed, and so on to the 
last point of the cord. Hence we conclude, that a system of 
forces applied in any manner at different points of a cord 
toill be' hi equilibrium, when, if applied at a single point 
without change of intensity or direction, they will maintain 
each other in equilibrium. 

Hence, we see that cords in machinery simply serve to 
transmit the action of forces, without in any other manner 
modifying their effects. 

The Lever. 

78. A lever is an inflexible bar, free to turn about an 
axis. This axis is called the fulcrum. 

Levers are divided into three classes, according to the 
relative positions of the points of application of the power 
and resistance. 

In the first class, the resistance is 
beyond both the power and fulcrum, 
and on the side of the fulcrum. The h , v 



common weighing-scale is an example i 

of this class of levers. The matter to p j, 

be weighed is the resistance, the « 

counterpoising weight is the power, F . g2 

and the axis of suspension is the 

fulcrum. 



ELEMENTARY MACHINES. 



99 



2nd Class. 



T 



E 

Fig. 63. 



3rd Class. 



JP 



J 



In the second class, the resistance 
is between the power and the ful- 
crum. The oar used in rowing a 
boat is an example of this class of 
levers. The end of the oar in the 
water is the fulcrum, the point at 
which the oar is fastened to the boat 
is the point of application of the resist- 
ance, and the remaining end of the oar 
i$ the point of application of the 
power. 

In the third class, the resistance is 
beyond both the fulcrum and the 
i power, and on the side of the power. 
The treadle of a lathe is an example 
of a lever of this kind. The point at 
which it is fastened to the floor is the 
fulcrum, the point at which the foot is 
applied is the point of application of 
the power, and the point where it is 

attached to the crank is the point of application of the 
resistance. 

Levers may be either curved or straight, and the direc- 
tions of the power and resistance may be either parallel or 
oblique to each other. We shall suppose the power and 
resistance to be situated in planes at right angles to the ful- 
crum ; for, if they were not so situated, we might conceive 
each to be resolved into two components— one at right 
angles, and the other parallel to the axis. The latter com- 
ponent would be exerted to bend the lever laterally, or to 
make it slide along the axis, developing only hurtful resist- 
ance, whilst the former only would tend to turn the lever 
about the fulcrum. 

The perpendicular distances from the fulcrum to the lines 
of direction of the power and resistance, are called the lever 
arms of these forces. In the bent lever MFN, the perpen« 




100 MECHANICS. 

dicular distances FA and FB are, respectively, the lever 
arms of P and R. 

To determine the conditions of 
equilibrium of the lever, let us 
denote the power by P, the re- 
sistance by JR, and their respec- 
tive lever arms by p and r. We 
have the case of a body restrained Fi<r 65 

by an axis, and if we take this as 

the axis of moments, we shall have for the condition of 
equilibrium (Art. 49), 

Pp = Pr; or, P ; M : : r : p . . ( 36.) 

That is, the power is to the resistance, as the lever arm of 
the resistance is to the lever arm of the power. 

This relation holds good for every kind of lever. 

The ratio of the power to the resistance when in equili 
brium, either statical or dynamical, is called the leverage, oi 
mechanical advantage. 

When the power is less than the resistance, there is said 
to be a gain of power, but a loss of velocity / that is, the 
space passed over by the power in performing any work, is 
as many times greater than that passed over b^y the resis- 
tance, as the resistance is greater than the power. When 
the power is greater than the resistance, there is said to be 
a loss of power, but a gain of velocity. When the power 
and resistance are equal, there is neither gain nor loss of 
power, but simply a change of direction. 

In levers of the first class, there may be either a gain or 
a loss of power ; in those of the second class, there is always 
a gain of power ; in those of the third class, there is always 
a loss of power. A gain of power is always attended with 
a corresponding loss of velocity, and the reverse. 

If several forces act upon a lever at different points, all 
being perpendicular to the direction of the fulcrum, they 
will be in equilibrium, when the algebraic sum of their 
moments, toith respect to the fulcrum, is equal to 0. 



ELEMENTARY MACHINES. 101 

This principle enables us to take into account the weight 
of the lever, which may be regarded as a vertical force 
applied at the centre of gravity. 

The pressure on the fulcrum is equal to the resultant of 
the power and resistance, together with the weight of the 
lever, when that is considered, and it may be found by the 
rule for finding the resultant of forces applied at points of a 
rigid body. 

The Compound Lever. 

79. A compound lever consists of a combination of 
simple levers AB, B C, CD, 

so arranged that the resis- ™ -g» 

tance in one acts as a power j T , ^ 

in the next, throughout the 



▼ 



P i rl P' 



' j J \r E- 



l b" i 



B 



combination. Thus, a power 
P produces at B a resis- 
tance B', which, in turn, 
produces at C a resistance Fig. 66. 

B", and so on. Let us as- 
sume the notation of the figure. From the principle of the 
simple lever, we shall have the relations, 

Pp = B'r", By = B"r\ B"p" = Br. 

Multiplying these equations together, member by member, 
and striking out the common factors, we have, 

Ppp'p" = Brr'r" ; or, P : B : : rr'r" : pp'p". ( Si.) 

We might proceed in a similar manner, were there any 
number of levers in the combination. 

Hence, in the compound lever, the power is to the resis- 
tance as the continued product of the alternate arms of 
lever, commencing at the resistance, is to the continued pro- 
duct of the alternate arms of lever, commencing at the 
power. 

By suitably adjusting the simple levers, any amount of 
mechanic il advantage may be obtained. 



102 MECHANICS. 

The following combination is used where a great pressure 
is to be exerted through a very small distance : 

The Elbow-joint Press. 
80. Let GA, BB, and BE represent bars, with hinge 
joints at B and B. The 

bar GA, has a fulcrum at A^ 

G, and the bar BE works /^^^^ b ,^ 

through a guide between / ^<§S^i>^ N * 

B and E. When A is ? f^~^_l^^^^ G 

depressed^ BE is forced [___ 

against the upright F, so Fig. 67. 

as to compress, with great 

force, any body placed between E and F. This machine is 

called the elbow-joint press, and is used in printing, in 

moulding bullets, in striking coins and medals, in punching 

holes, riveting steam boilers, &c. 

Let P denote the force applied at A, perpendicular to 
AG, Q the resistance in the direction BB, and R the com- 
ponent of Q, in the direction EB. Let G be taken as an 
axis of moments, and then, because P and Q are in equili- 
brium, we shall have, 

P X AG = Q x EG, or, Q = P x ~ 

If we draw BH perpendicular to BR, we shall have, 

T)JT 

cos BBH = yy^ ; but we have, for the component R, 

T)TT 
R = QcosBBH= Q x -^- 

Substituting for Q its value, and reducing, 

R _ AG BH 

P - EG X BB ' 

When B is depressed, BIT and BB approach equality, 
and EG continually diminishes ; that is, the mechanical ad- 
vantage increases, and finally, when B reaches ER, it 
becomes infinite. There is no limit to the pressure exerted 
at F, except that fixed by the strength of the machine. 



ELEMENTARY MACHINES. 



10S 



G^SiK 




Fig. 68. 



Tne Balance. 

aii A Balance is a machine for weighing bodies: it 
consists of a lever AB, called the 
beam, a knife edge fulcrum F, and 
two scale-pan^ D and F, suspended 
by knife-edges from the extremities 
of the lever arms FB and FA. 
These arms should be symmetrical, 
and of equal length ; the knife- 
edges A, B, and F, should all lie 
in the same plane, and be perpen- 
dicular to a plane through their 

middle points and the centre of gravity of the beam ; they 
are, therefore, parallel to each other. This condition of 
parallelism in the same plane, is of essential importance. 

In addition to this, the middle points of the knife-edges A, 
B, and F, should be on the same straight line, perpendicular 
to the plane through the fulcrum F, and the centre of gravity 
of the beam. The knife-edges should be of hardened steel, 
and their supports should either be of polished agate, or, 
what is still better, of hardened steel, so as to diminish the 
eifect of friction along the lines of contact. The fulcrum 
may be made horizontal, by leveling-screws passing through 
the foot-plate L. A needle JV, projects upwards, or some- 
times downwards, which, playing in front of a graduated 
arc Gil, serves to show the deflection of the line of knife- 
edges from the horizontal. When the instrument is not in 
use, the fulcrum may be raised from its bearings by a pinion 
K, working into a rack in the interior of the standard FK. 
The knife-edges A and 3 may, by a similar arrangement, 
be raised from their bearings also. 

The ordinary balances of the shops are similar in their 
general plan ; but many of the preceding arrangements are 
omitted. The scale-pans being exactly alike, the balance 
should remain in equilibrium, with the line AB horizontal, 
not only when the balance is without a load, but also when 
the pans are loaded with equal weights ; and when AB is 



104: MECHANICS. 

deflected from the horizontal, it should return to this posi- 
tion. This result is attained by throwing the centre of 
gravity slightly below the line AB. To test a balance, let 
two weights be placed in the pans that will exactly counter- 
balance each other, then change the weights to the opposite 
pans ; if the equilibrium is still maintained, the balance is 
said to be true. 

The sensibility of a balance is its capability of indicating 
small differences of weight. The sensibility will be greater, 
as the lengths of the ar?ns increase, as the centre of gravity 
of the beam approaches the fulcrum, as the mass of the 
load decreases, and as the length of the needle increases. 
The centre of gravity of the beam being below the fulcrum, 
it may be made to approach to or recede from it, by a solid 
ball of metal attached to the beam by means of a screw, by 
which it may be raised or depressed at pleasure. The 
remaining conditions of sensibility will be limited by the 
strength of the material, and the use to whkm it is to be 
applied. 

Should it be found that a balance is not true, it may still 
be employed, with but slight error, as indicated below. 

Denote the length of the lever arms, by r and r\ and the 
weight of the body, by W. When the weight W is applied 
at the extremity of the arm r, denote the counterpoising 
weights employed, by W ; and when it is applied at the 
extremity of the arm r', denote the counterpoising weights 
employed, by W" . We shall have, from the principle of the 
lever, 

Wr =z Wr', and Wr' = W"r. 

Multiplying these equations, member by member, we have, 

Wrr' = W'Wrr' ; /. W= <y/W'W h \ 

that is, the true weight is equal to the square root of the pro- 
duct of the apparent weights. 

A still better method, and one that is more free from the 
effects of errors in construction, is to place the body to be 




Fig. 69. 



ELEMENTARY MACHINES. 105 

weighed in one scale and add counterpoising weights till the 
beam is horizontal ; then remove the body to be weighed 
and replace it by known weights till the beam is again hori- 
zontal ; the sum of the replacing weights will be the weight 
required. If, in changing the loads, the positions of the 
knife-edges are not moved, this method is almost exact, but 
this is a condition difficult to fulfill in manipulation. 

The Steelyard. 

82. The steelyard is an instrument used for weighing 
bodies. It consists of a lever AB, called the beam ; a ful- 
crum F\ a scale-pan J9, 
attached at the extremity 
of one arm; and a known 
weight E, movable along 
the other arm. We shall 
suppose the weight of E to 
be 1 lb. Tins instrument 
is sometimes more conve- 
nient than the balance, but it is more inaccurate. The con- 
ditions of sensibility are essentially the same as for the 
balance. To graduate the instrument, place a pound-Aveight 
in the pan _Z>, and move the counterpoise E till the beam 
rests horizontal — let that point be marked 1 ; next place a 
10 lb. weight in the pan, and move the counterpoise E till 
the beam is again horizontal, and let that point be marked 
10; divide the intermediate space into nine equal parts, and 
mark the points of division as shown in the figure. These 
spaces may be subdivided at pleasure, and the scale ex- 
tended to any desirable limits. We have supposed that the 
centre of gravity coincides with the fulcrum ; when this is 
not the case, the weight of the instrument must be taken 
into account as a force applied at its centre of gravity. We 
may then graduate the beam by experiment, or we may 
compute the lever arms, corresponding to the different 
weights, by the general principle of moments. 

To weigh any body with the steelyard, place it in the 
scale-pan and move the counterpoise E along the beam til] 
5* 



106 



MECHANICS. 



an equilibrium is established between the two ; the cor 
responding mark on the beam will indicate the weight. 

The bent Lever Balance. 



This balance consists of a bent lever A GB ; 
C ; a scale-pan 




83. 

fulcrum 

D; and a graduated arc 
EF, whose centre co- 
incides with the centre 
of motion C. When a 
weight is placed in the 
scale-pan, the pan is de- 
pressed and the lever- 
arm of the weight is 

diminished ; the weight B is raised, and its lever-arm 
increased. When the moments of the two forces become 
equal, the instrument will come to a state of rest, and the 
weight will be indicated by a needle projecting from B, and 
playing in front of the arc FE. The zero of the arc EF is 
at the point indicated by the needle when there is no load in 
the pan J). 

The instrument may be graduated experimentally by 
placing weights of 1, 2, 3, <fcc, pounds in the pan, and mark- 
ing the points at which the needle comes to rest, or it may 
be graduated by means of the general principle of moments. 
We need not explain this method of graduation. 

To weigh a body with the bent lever balance, place it in 
the scale-pan, and note the point at which the needle comes 
to rest ; the reading will make known the weight sought. 



Compound Balances. 

§4. Compound balances are much used in weighing 
heavy articles, as merchandise, coal, freight for shipping, 
&c. A great variety of combinations have been employed, 
one of which is annexed. 

AB is a platform, on which the object to be weighed is 



ELEMENTARY MACHINES. 



107 




Fig. 71. 



placed; BG is a guard 
firmly attached to the 
platform ; the platform 
is supported upon the 
knife-edge fulcrum E, 
and the piece D, through 
the medium of a brace 
CD ; GEis a lever turn- 
ing about the fulcrum E, 
and suspended by a rod from the point L ; LN' is a lever 
having its fulcrum at Ji", and sustaining the piece D by a 
rod KH\ is a scale-pan suspended from the end iVof the 
lever LN. The instrument is so constructed, that 

EF ': GE: : KM: LM; 

and the distance EM is generally made equal to y 1 ^ of 3IN. 
The parts are so arranged that the beam LN shall rest 
horizontally in equilibrium when no weight is placed on the 
platform. 

If, now, a body Q be placed upon the platform, a part of 
its weight will be thrown upon the piece Z>, and, acting 
downwards, will produce an equal pressure at E. The 
remaining part will be thrown upon E, and, acting upon the 
lever EG, will produce a downward pressure at G, which 
will be transmitted to L ; but, on account of the relation 
given by the above proportion, the effect of this pressure 
upon the lever LN will be the same as though the pressure 
thrown upon E had been applied directly At E. The final 
effect is, therefore, the same as though the weight of Q had 
been applied at IT, and, to counterbalance it, a weight equal 
to T L of Q must be placed in the scale-pan 0. 

To weigh a body, then, by means of this scale, place it on 
the platform, and add weights to the scale-pan till the lever 
LN is horizontal, then 10 times the sum of the weight 
added will be equal to the weight required. By making 
other combinations of levers, or by combining the princi- 



108 MECHANICS. 

pie of the steelyard with this balance, objects may be 
weighed by using a constant counterpoise. 

EXAMPLES. 

1. In a lever of the first class, the lever arm of the 
resistance is 2§ inches, that of the power, 33^, and the 
resistance 100 lbs. What is the power necessary to hold 
the resistance in equilibrium ? Ans. 8 lbs. 

2. Four weights of 1, 3, 5, and 7 lbs. respectively, are 
suspended from points of a straight lever, eight inches apart. 
How far from the point of application of the first weight 
must the fulcrum be situated, that the weights may be in 
equilibrium ? 

SOLUTION. 

Let x denote the required distance. Then, from Art. (36) 
1 X x + S(x - 8) -h 5(x - 16) + 1(x — 24) = ; 
.*. x = 17 in. Ans. 

3. A lever, of uniform thickness, and 12 feet long, is 
kept horizontal by a weight of 100 lbs. applied at one 
extremity, and a force JP applied at the other extremity, so 
as to make an angle of 30° with the horizon. The fulcrum 
is 20 inches from the point of application of the weight, and 
the weight of the lever is 10 lbs. What is the value of JP, 
and what is the pressure upon the fulcrum ? 

SOLUTION. 

The lever arm- of JP is equal to 124 in. X sin 30° = 62 in., 
and the lever arm of the weight of the lever is 52 in. 
Hence, 
20 x 100 = 10 X 52 + JP X 62 ; .'. JP = 24 lbs. nearly. 

We have, also, 



R = -v/JT 2 + T 2 = ^/(U0 + 24 sin 30°) 2 -h (24cos30°) 9 . 
.-. R = 123.8 lbs. : 



ELEMENTARY MACHINES. 109 

X 20.785 
and, cos« = -g = ^^- = .16789; 

.-. a = 80° 20' 02". 

4. A aeavy lever rests on a fulcrum which is 2 feet from 
one end, 8 feet from the other, and is kept horizontal by a 
weight of 100 lbs., applied at the first end, and a weight 
of 18 lbs., applied at the other end. What is the weight 
of the lever, supposed of uniform thickness throughout ? 

SOLUTION. 

Denote the required weight by x; its arm of lever is 
3 feet. We have, from the principle of the lever, 

100 x 2 = x x 3 + 18 X 8 ; .*. as = 18f lbs. Arts. 

5. Two weights keep a horizontal lever at rest ; the 
pressure on the fulcrum is 10 lbs., the difference of the 
weights is 4 lbs., and the difference of lever arms is 9 inches. 
What are the weights, and their lever arms ? 

Ans. The weights are 7 lbs. and 3 lbs. ; their lever arms 
are 15| in., and 6f in. 

6. The apparent weight of a body weighed in one pan 
of a false balance is b\ lbs., and in the other pan it is 
6 T 6 T lbs. What is the true weight ? 



W= yy x f f .= 6 lbs. Ans. 

7. In the preceding example, what is the ratio of the 
lever arms of the balance ? 

SOLUTION. 

Denote the shorter arm by £, and the longer arm by nU 
We shall have, from the principle of moments, 

6£ = 5J X nl, or, 6nl = 6 T 6 T Z ; .*. n ~ \ T \. 

That is, the longer arm equals 1 T \ times the shorter arm 




110 MECHANICS. 

The Inclined Plane. 

85. An inclined plane is a plane inclined to the horizon, 
In this machine, let the power be a force applied to a body 
either to prevent motion down the plane, or to produce 
motion up the plane, and let the resistance be the weight of 
the body acting vertically downwards. The power may be 
applied in any direction whatever ; but we shall, for sim- 
plicity's sake, suppose it to be in a vertical plane, taken per- 
pendicular to the inclined plane. 

Let AB represent the inclined plane, a body resting 
on it, B the weight of the body, 
and P the force applied to hold it 
in equilibrium. In order that these 
two forces may keep the body at 
rest, friction being neglected, their 
resultant must be perpendicular to 
AJB (Art. 12). Fig-72 . 

When the direction of the force 
P is given, its intensity may be found geometrically, as fol- 
lows : draw OB to represent the weight, and Q perpen- 
dicular to AB ; through B draw B Q parallel to OB, and 
through Q draw QP parallel to OB ; then will OP repre- 
sent the required intensity, and OQ the pressure on the 
plane. 

When the intensity of P is given, its direction may be 
found as follows : draw OB and Q as before ; with B as 
a centre, and the given intensity as a radius, describe an 
arc cutting OQ in Q\ draw BQ, and through draw OP 
parallel, and equal to B Q ; it will represent the direction 
of the force P. 

If we denote the angle between P and B by p, and the 
inclination of the plane by a, we shall have the angle BOQ 
equal to a, since Q is perpendicular to AB, and OB to 
A (7, and, consequently, the angle Q OP = 9 — a s From 
the principle of Art. 35, we have, 

P : B : : sina : sinfa — a) . . ( 38.) 



ELEMENTARY MACHINES. 



Ill 



From which, if either P or <p be given, the other can be 
found. 

If we suppose the power to be 
applied parallel to the plane, we 
shall have, 9 — a =90°, 

or, sin(<p — a) = 1. 

We have, also, sur« = |g. 

AB 

Substituting these in the preced- 
ing proportion, and reducing, we 
have, 

P : B : : BG : AB 




Fig. 73. 



(39.] 



That is, when the power is parallel to the plane, the power 
is to the resistance, as the height of the plane is to its length. 

If the power is parallel to the base of the plane, we shall 
have, <p — a = 90° — a ; whence, 



sin((p — a) z= cos a = 



AC, 
AB' 



also, 



sin a 



BC 
AB 




Substituting in Proportion (38), 
and reducing, we have, 

P : R :: BC : AG 



Fig. 74. 



(40. 



That is, the power is to the resistance as the height of the 
plane is to its base. 

From the last proportion we have, 

P = B^ = iftana. 
AC 

If we suppose a to increase, the value of P will increase, 
and when a becomes 90°, P will become infinite ; that is, if 
friction be neglected, no finite horizontal force can sustain a 
body against a vertical wall. 



112 



MECHANICS. 



EXAMPLES. 



1. A power of 1 lb., acting parallel to an inclined plane, 
supports a weight of 2 lbs. What is the inclination of the 
plane? Ans. 30°. 

2. The power, resistance, and normal pressure, in the 
case of an inclined plane, are, respectively, 9, 13, and 6 lbs. 
What is the inclination of the plane, and what angle does 
the power make with the plane ? 

SOLUTION. 

If we denote the angle between the power and resistance 
by <p, and the inclination of the plane by a, we shall have, 
from Art. (35), 

6 = VIZ* + 9 a + 2 X 9 X 13 cos <p ; 

/. <p = 156° 8' 20". 

Also, from Art. (35), for the inclination of the plane, 

6 : 9 : : sin 156° 8' 20" : sin a ; .-. a = 37° 21' 26". 

Inclination of power to plane — cp — 90° — a = 28° 46' 54". 

Ans. 

3. A body may be supported on an inclined plane by a 
force of 10 lbs., acting parallel to the plane ; but it requires 
a force of 12 lbs. to support it when the force acts parallel 
to the base. What is the weight of the body, and what is 
the inclination of the plane ? 

Ans. The weight is 18.09 lbs., and the inclination is 
33° 33' 25". 

The Pulley. 

86. A pulley consists of a wheel having a groove around 
its circumference to receive a cord ; the wheel turns freely 
on an axis at right angles to its plane, which axis is sup« 
ported by a frame called a block. The pulley is said to bo 
fixed, when the block is fixed, and to be movable, when 



ELEMENTARY MACHINES. 



11: 



the block is movable. Pulleys may be used singly, or in 

combinations. 

Single fixed Pulley. 

87. In this pulley the block, and, consequently, the axis, 
is fixed. Denote the power by P, the resist- 
ance by J?, and the radius of the pulley by r. 
It is plain that both the power and resistance 
should be in a plane, at right angles to the 
axis. Hence, if we take the axis of the pulley 
as the axis of moments, we shall have (Art. 49), 
the following condition of equilibrium : 



Pr = Br; or, P 



B. 



rop, 



p s 

Fig. 75. 



That is, in the single fixed pulley, the poicer is equal to the 
resistance. 

The effect of the pulley is, therefore, simply to change the 
direction of the force, and it is for this purpose that it is 
generally used. 

Single Movable Pulley. 

88. In this pulley the block, and, consequently, the 
axis, is movable. The resistance is applied at 
a hook attached to the block; one end of a 
rope, enveloping the lower part of the pulley, 
is firmly attached at a fixed point (7, and the 
power is applied at the other extremity. We 
shall take the two branches of the rope par- 
allel, that being the most advantageous way of 
using the machine. 

Adopting the notation of the preceding 
article, and taking A, the point of contact of 
CA with the pulley, as the centre of moments, 
we shall have, for the condition of equilibrium 
(Art. 49), 




P x 2r = Br ; 



P = ±B. 



That is, in the movable pulley, when the power and 
resistance are parallel, the poicer is equal to one half of tht 
resistance. The tension upon the cord CA is evidently the 



1U 



MECHANICS. 



same as that upon the cord BP. It is, therefore, equal to 
the power, or to one-half the resistance. If, therefore, 
the resistance of the fixed point C be replaced by a force 
equal to P, the equilibrium will be undisturbed. 

If the two branches of the en- 
veloping cord are oblique to each 
other, the condition of equilibrium 
will be somewhat modified. Sup- 
pose the resistance of the fixed 
point C to be replaced by a force 
equal to P, and denote the angle 
between the two branches of the 
cord by 2p. If an equilibrium 
subsists between the forces P, P, 
and P, we must have the relation, 




Fig. 11. 



2 Pcosp = B. 

Draw the chord AB between the points of contact of the 
cord and pulley, and denote its length by c ; draw, also, 
the radius OB. Then, since OB is perpendicular to AB 
and BP to OB, the angle AB will be equal to one half 
of the angle A CB, or equal to 9. Hence, 



coscp 



\c -f- r = — - 
2 2r 



Substituting in the preceding equation and reducing, we 
have, 

Pc = Br; .-. P : B : r : c . . (41.) 

That is, the power is to the resistance as the radius of the 
pulley is to the chord of the arc enveloped by the rope. 

When the chord is greater than the radius, there will 
be a gain of mechanical advantage in the use of this pulley ; 
when less, there will be a loss of mechanical advantage. 

If the chord becomes equal to the diameter, we have, as 

before, 

P ±- Ut\ 



ELEMENTA K Y MACHINES. 



115 



Combinations of Separate Movable Pulleys. 

89. The figure represents a combination of three movable 
pulleys, in which there are as many separate 
cords as there are pulleys ; the first end of 
each cord is attached at a fixed point, the 
second end being fastened to the hook of 
the next pulley in order, except the last 
cord, at the second extremity of which the 
power is applied. 

Let us denote the tension of the cord 
between the first and second pulley by t, 
that of the cord between the second and 
third pulley by t'. By the preceding 
Article, we have, 




Fig. 78. 



t = \R; t' = it; P = %t f . 



Multiplying these equations together, member by member, 
and striking out the common factors in the resulting equa- 
tion, we have, 

p = ays. 

Had there been n pulleys in the combination, we should 
have obtained, in an entirely similar manner, the relation, 



(i)*-£; 



P: B 



1 



(42.) 



That is, the power is to the resistance as 1 is to 2", n 
denoting the number of pulleys. 

For convenience, the last branch of the cord is often 
passed over a fixed pulley ; this arrangement only serves to 
change the direction of the force, without in any way chang- 
ing the conditions of equilibrium. 

Combinations of Pulleys in blocks. 

90. These combinations are effected in a variety of 

ways. In most cases, there is but a single rope employed, 

which, being firmly attached to a hook of one block, passes 

around a pulley hi the other block, then around one in the 



1 1 6 MECHANICS. 

first block, and so on, passing from block to block until it 
has passed around each pulley in the system. The power is 
applied at the free end of the rope. Sometimes the pulleys 
in the same block are placed side by side, sometimes they 
are placed one above another, as represented in the figure, 
in which case the interior pullies are made 
somewhat smaller than the outer ones. The 
conditions of equilibrium are the same in both 
cases. To deduce the conditions of equili- 
brium in the case represented, in which the 
upper block is fixed and the lower one mov- 
able : denote the power by i°, the resistance ( 
by J?. When there is an equilibrium between 32 
jP and i£, the tension upon each branch of 
the rope which aids in supporting the resist- 
ance must be the same, and equal to P; but, 
since the last pulley simply serves to change X 

the direction of the force J°, there will be -^ 

four such branches in the case considered ; Fi T9 

hence, we shall have, 

4P = B, or P = ±R. 

Had there been n pulleys in the combination, there would 
have been n supporting branches of the cord, and we should 
have had, in the same manner, 

nP=z JR, or P : R : : 1 : n . . ( 43.) 

That is, the power is to the resistance as 1 is to the num- 
ber of branches of the rope which support the resistance. 

The principles involved in the combinations already con- 
sidered, will be suincient to make known the relation 
between the power and resistance in any combination what- 
ever. 

EXAMPLES. 

1. In a system of six movable pulleys, of the kind des- 
cribed in Art. 89, what weight can be sustained by a power 
of 12 lbs? Am. 768 lbs. 



ELEMENTARY MACHINES. 



117 



2. Li a combination of pulleys in two blocks, when 
there are six pulleys in each block, what weight can a power 
of 12 lbs. sustain in equilibrium? Ans. 144 lbs. 

3. In a combination of separate movable pulleys, the 
resistance is 576 lbs., and the power which keeps it in equi- 
librium is 9 lbs. How many pulleys are there in the com- 
bination ? A?is. 6. 

4. In a combination of pulleys in two blocks, with a single 
rope, the power is 62 lbs., and the resistance 496 lbs. How 
many pulleys are there in each block ? Ans. 4. 

5. In a combination of two movable pulleys, the inclina- 
tions of the ropes at each pulley is 120°. What is the power 
required to support a weight of 27 lbs. ? Ans. 9 lbs. 

The Wheel and Axle. 

91. The wheel and axle consists of a wheel A, mounted 
on an axle or arbor B. The power is 
applied at one extremity of a rope 
wrapped around the wheel, and the 
resistance at one extremity of a sec- 
ond rope, wrapped around the axle in 
a contrary direction. The whole in- 
strument is supported by pivots pro- 
jecting from the ends of the axle. In 
deducing the conditions of equili- 
brium of the power and resistance, we shall suppose them tc 
be situated in planes, at right angles to the axis. 

Denote the power by P, the re- 
sistance by i?, the radius of the 
wheel by r, and the radius of the 
axle by r '. We shall have, in case 
of an equilibrium (Art. 49), 




F:g. SO. 



Pr=Rr', orP : R : : r' 



(44.) 



That is, the power is to the resistance 
as the radius of the axle is to the 
r-i<7hrs of the wheel. 




U8 



MECHANICS. 



By suitably varying the dimensions of the wheel and axle, 
any amount of mechanical advantage may be obtained. 

If we draw a straight line from the point of contact of 
the first rope and the wheel, to the point of contact of the 
second rope and the axle, the power and resistance being 
parallel, it can readily be shown that it will cut the axis of 
revolution at a point which divides the line through the 
points of contact into two parts, which are inversely pro- 
portional to the power and resistance. Hence, this is the 
point of application of the resultant of these two forces. 
The resultant will be equal to the sum of the forces, and by 
the aid of the principle of moments, the pressure on each 
pivot may be computed. When the weight of the machine 
is to be taken into account, we must regard it as a vertical 
force applied at the centre of gravity of the wheel and axle. 
The pressures upon each pivot due to this weight, may be 
computed separately, and added to those already found. 

Combinations of Wheels and Axles. 

92. If the rope of the first axle be passed around a 
second wheel, and the rope of the second axle around a 
third wheel, and so on, a combination will result which is 
capable of affording great mechanical advantage. The 
figure represents a combination of two 
wheels and axles. To deduce the 
conditions of equilibrium, denote the 
power by _P, the resistance by i?, the 
radius of the first wheel by r, that of 
the first axle by r', that of the second 
wheel by r", and that of the second 
axle by r" r . If we denote the tension 
of the connecting rope by £, this may 
be regarded as a power applied to the 
second wheel. From what was de- 
monstrated for the wheel and axle, we 
shall have, 

Pr = tr\ and tr" = Br' 




ELEMENTARY MACHINES. 



119 



Multiplying these equations together, member by member, 
and reducing, we have, 

Prr" = Pr'r'" ; or, P : JR : : r'r'" 



rr . 



In like manner, were there any number of wheels and 
axles in the combination, we might deduce the relation, 



or, 



Prr"r lv . . 
P : R : : rV"V v 



ifrW 



(45.) 



That is, the power is to the resistance as the continued 
product of the radii of the axles is to the continued product 
of the radii of the wheels. 

The principle just explained, is applicable to those kinds 
of machinery in which motion is transmitted from wheel to 
wheel by the aid of bands, or belts. An endless band, 
called the driving belt, passes around one drum mounted 
upon the axle of the driving wheel, and around another on 
that of the driven wheel. When the radius of the former 
is greater than that of the latter, there is a gain of velocity, 
and a corresponding loss of power ; in the contrary case, 
there is a loss of velocity, and a corresponding gain of 
power. In the first case, we are said to gear up for velo- 
city / in the second case, we are said to gear down for 
power. These remarks admit of extension to combinations 
of any number of pieces, in which motion is transmitted b} 
belts, cords, chains, or, as we shall see hereafter, by trains 
of toothed wheels. 



93. 



The Crank and Axle, or Windlass. 

This machine con- 



sists of an axle AP, and a 
crank B CD. The power 
is applied to the crank-han- 
dle D C, and the resistance 
to a rope wrapped around 
the axle. The distance from 
the handle PC to the axis, 
is called the crank-arm. 




Fig. 88. 



120 



MECHANICS. 



The relation between the power and resistance, when in 
equilibrium, is the same as in the wheel and axle, except 
that we substitute the crank-arm for the radius of the wheel. 

Hence, the power is to the resistance as the radius of the 
axle is to the crank-arm. 

This machine is used in drawing water from wells, raising 
ore from mines, and the like. It is also used in combina- 
tion with other machines. Instead of the crank, as shown 
in the figure, two holes are sometimes bored at right angles 
to each other and to the axis, and levers inserted, at the ex- 
tremities of which the power is applied. The condition of 
equilibrium remains unchanged, provided we substitute for 
the crank-arm, the distance from the point of application of 
the power to the axis. 

The Capstan. 

94. The Capstan differs in no material respect from the 
windlass, except in having its axis vertical. The capstan 
consists of a vertical axle passing through strong guides, 
and having holes at its upper end for the insertion of levers. 
It is much used on shipboard for raising anchors. The con- 
ditions of equilibrium are the same as in the windlass. 

The Differential Windlass. 

95. This differs from the common windlass in having an 
axle formed of two cylinders, 

A and B, of different dia- 
meters, but having a com- 
mon axis. A rope is attached 
to the larger cylinder, and 
wrapped several times around 
it, after which it passes around 
the movable pulley C, and, 
returning, is wrapped in a 
contrary direction about the 
smaller cylinder, to which the 
second end of the rope is 
made fast. The power is ap- 
plied at the crank-handle FE, and the resistance to the hook 
of the movable pulley. When the crank is turned so as to 




ELEMENTARY MACHINES. 121 

wind the rope upon the larger cylinder it unwinds from 
the smaller one, but in a less degree, and the total effect of 
the power is to raise the resistance JR. To deduce the 
conditions of equilibrium between the power and resistance, 
denote the power by P, the resistance by P, the crank-arm 
by c, the radius of the larger cylinder by r, and that of the 
smaller cylinder by r'. The resistance acts equally upon 
the two branches of the rope from which it is suspended, 
hence the tension of each branch may be represented by 
AJR. Suppose that the pow T er acts to wind the rope upon 
the larger cylinder. The moment of the power will be 
Pc ; the moment of the tension of the branch A will be equal 
to \Mr\ this acts to assist the power ; the moment of the 
tension of the branch JB will be equal to \Mr, this acts to op- 
pose the power. From the principle of moments, we have, 

Pc + \Rr' = \Pr, or Pc = JP (r - r') ; 
whence, 

P : M : : r — r' : 2c. . . . ( 46.) 

That is, the power is to the resistance as the difference of 
the radii of the two cylinders is to twice the crank-arm. 

By increasing the crank-arm and diminishing the differ- 
ence between the radii of the cylinders, any amount of 
mechanical advantage may be obtained by the use of this 
machine. 

Wheel-work. 

96. The principle employed in finding the relation 
between the power and resist- 
ance in a train of wbeel-work 
is the same as that used in 
discussing the wheel and axle 
and its modifications. To illus- 
tratc the method of proceed- 
ing, w^e have taken the case in 
which the power is applied to a 
crank-handle which is attached 
to the axis of a cogged wheel Fig ' ** 

6 




122 



MECHANICS. 




A ; the teeth, or cogs, of this wheel work into the spaces 
of the toothed wheel B, and 
the resistance is attached to a 
rope wound round the arbor 
of the last wheel. In order 
that the wheel A may com- 
municate motion freely to the 
wheel B, the number of teeth 
in their circumferences should 
be proportional to their radii, 
and the spaces between the 
teeth in one wheel should be large enough to receive the 
teeth of the other wheel, but not large enough to allow a 
great deal of play. The teeth should always come in contact 
at the same distances from the centres of the wheels, and 
those distances are taken as the radii of the wheels them- 
selves. Denote the power by P, the resistance by B, the 
crank-arm by c, the radius of the wheel A by r, that of the 
wheel B by r\ that of the arbor by r", and suppose the 
power and resistance to be in equilibrium ; then will the 
pressure due to the action of the power tend to turn the 
wheels in the direction of the arrow heads. This tendency 
will be counteracted by the pressure of the resistance tend- 
ing to produce motion in a contrary direction. If we 
denote the pressure at the point C by B\ we should have, 
from what has preceded, 

Pc = B'r and BY = Br" ; 

whence, by multiplication and reduction, 

Per' = Brr", or P : B : : rr" : cr' . ( 47.) 



That is, the power is to the resistance as the continued 
product of the alternate arms of lever, beginning at the 
resistance, is to the continued product of the alternate arms 
of lever beginning at the power. 

Had there been any number of wheels in the train lying 



ELEMENTARY MACHINES. 123 

between the power and resistance, we should have found 
similar conditions of equilibrium. 

EXAMPLES . 

1. A power of 5 lbs., acting at the circumference of a 
wheel whose radius is 5 feet, supports a resistance of 200 lbs. 
applied at the circumference of the axle. What is the 
radius of the axle ? Ans. \\ inches. 

2. The radius of the axle of a windlass is 3 inches, and 
the crank-arm 15 inches. What power must be applied to 
the crank-handle, to support a resistance of 180 lbs., applied 
to the circumference of the axle ? Ans. 36 lbs. 

3. A power P, acts upon a rope 2 inches in diameter, 

passhig over a wheel whose radius is 3 feet, and supports a 

resistance of 320 lbs., applied by a rope of the same diame 

ter, passing over an axle whose radius is 4 inches. What is 

the value of .P, when the thickness of the rope is taken into 

account. Ans. 43g 9 y lbs. 

The Screw. 

97. The screw is essentially a combination of two in 
clined planes. It consists of a solid cylinder, 
called the cylinder of the screw, which is en- 
veloped by a spiral projection called the 
thread. The thread may be generated as 
follows : let an isosceles triangle be placed so 
that its base shall coincide with an element of 
the cylinder of the screw, and so that its 
plane shall pass through the axis. Let the Fig. 86. 

triangle be revolved uniformly about the axis, 
and at the same time be moved uniformly in the direction 
of the axis, at such a rate that it shall pass over a distance 
in this direction equal to the base of the triangle during one 
revolution. The solid generated by the triangle is the 
thread of the screw. The two sides of the triangle generate 
helicoidal surfaces, which constitute the upper and lower 
surfaces of the thread. Every point in these lines generates 
a curve called a helix, which is entirelv similar to an inclined 




124 MECHANICS. 

plane bent around a cylinder. The vertex generates what 
is called the outer helix, and the two angular points of th6 
base trace out the same curve, which is the inner helix. 
The screw just described is called a screw with a triangular 
thread. Had we used a rectangle, instead of a triangle, and 
imposed the condition, that the motion in the direction of 
the axis during one revolution, should be equal to twice the 
base, we should have had a screw with a rectangular thread, 
as in the figure. 

The screw works into a piece called a nut, which is gene- 
rated in a manner entirely analogous to that just described, 
except that what is solid in the screw is wanting in the nut ; 
it is, therefore, exactly adapted to receive the thread of the 
screw. Sometimes, the screw remains fast, and the nut is 
turned upon it ; in which case, the nut has a motion of revo- 
lution, combined with a longitudinal motion. Sometimes, 
the nut remains fast, and the screw is turned within it, in 
which case, the screw receives a motion in the direction of 
its axis, in connection with a motion of rotation. The con- 
ditions of equilibrium are the same for each. In both cases, 
the power is applied at the extremity of a lever ; when in 
motion, the point of application describes an ascending or 
descending spiral, resulting from a combination of the 
rotary and the longitudinal motion We shall suppose the 
nut to remain fast, and the screw to be movable, and that 
the resistance acts parallel to the axis of the screw. If the 
axis is vertical, and the resistance a weight, we may regard 
that weight as resting upon one of the helices, and sustained 
in equilibrium by a force applied horizontally. If we suppose 
the supporting helix to be developed on a vertical plane, it 
will form an inclined plane, whose base is the circumference 
of the base of the cylinder on which it lies, and whose alti- 
tude is the distance between the threads of the screw. 

Let AB represent the development of this helix on a 
vertical plane, and denote by F the force applied parallel to 
the base, and immediately to the weight R, to sustain it 
on the plane. We shall have (Art 85), 

F: R :: BC : AC. 




ELEMENTARY MACHINES. 125 

But the power is actually- 
applied through the medium 
of a lever. Denoting the 
radius OG of the cylinder of 
the supporting helix, by r, and 
the arm of lever of the power 
P by p, we shall have, from 
the principle of the lever, 

P : JP : : r : p ; Fig- 87. 

or, P : F : : 2«r : 2«p. 

Combining this proportion with the preceding one, and 
recollecting that AG — 2tfr, we deduce the proportion, 

P : R : : BG : 2*p . . . . (48.) 

That is, the power is to the resistance as the distance be- 
tween the threads is to the circumference described by the 
point of application of the power. 

By suitably diminishing the distance between the threads, 
other things being equal, any amount of mechanical ad- 
vantage may be obtained. 

The screw is used for producing great pressures 
through very small distances, as in pressing books for the 
binder, packing merchandise, expressing oils, and the like. 
On account of the great amount of friction, and other hurt- 
ful resistances developed, the modulus of the machine is 

very small. 

The Differential Screw. 

98. The differential screw consists essentially of an ordi- 
nary screw, as just described, into the end of which works 
a smaller screw, having its axis coincident with the first, 
but having its thread turned in a contrary direction ; that 
is, it is what is technically called a left-handed screw, the 
first screw being a right-handed one. The distance between 
the threads of the second screw is somewhat less than that 
between the threads of the first screw, and this difference 



126 



MECHANICS. 



may be made as small as desirable. The second screw is su 
arranged that it admits of a longitudinal motion, but not of 
a motion of rotation. By the action of the differential screw, 
the weight is raised vertically through a distance equal to 
the difference of the distances between the threads on the 
two screws, for each revolution of the point of application 
of the power. For, were the first screw alone to turn, the 
weight would be raised through a distance equal to the dis- 
tance between its threads ; but, because the second screw is 
a left-handed one, this distance will be diminished by a dis- 
tance equal to that between its threads. We may, there- 
fore, write the following rule : 

The power is to the resistance as the difference of the 
distances between the threads of the two screws is to the cir- 
cumference described by the point of application of the 
power. 

Endless Screw. 

99. The endless screw is a screw secured by shoulders, 
so that it cannot be moved longi- 
tudinally, and working into a 
toothed wheel. The distance be- 
tween the teeth should be nearly 
equal to the distance between 
the threads of the screw. When 
the screw is turned, it imparts a 
rotary motion to the wheel, which 
may be utilized by any mechani- 
cal device. The conditions of 
equilibrium are the same as for 
the screw, the resistance in this 
case being offered by the wheel, 
in the direction of its circumference. 

Machines of this kind are used in determining the number 
of revolutions of an axis. An endless screw is arranged to 
turn as many times as the axis, and being connected with a 
train of light wheel-work, the last piece of which bears an 
index, the number of revolutions can readily be ascertained 




Fig. 88. 



ELEMENTARY MACHINES. 127 

at any instant. As an example, suppose the first wheel to 
have 100 teeth, and to bear on its arbor a smaller wheel, 
having 10 teeth; suppose this wheel to engage with a larger 
wheel having 100 teeth, and so on. When the endless 
screw has made 10,000 revolutions, the first wheel will have 
made 100 revolutions, the second large wheel will have 
made 10 revolutions, and the third wheel 1 revolution. By 
a suitable arrangement of indices, the exact number of revo- 
lutions of the axis, at any instant, may be read off from the 
instrument. 

EXAMPLES. 

1. What must be the distance between the threads of a 
screw in order that a power of 28 lbs., acting at the ex- 
tremity of a lever 25 inches long, may sustain a weight of 
10,000 lbs.? Arts. .4396 inches. 

2. The distance between the threads of a screw is ^ of an 
inch. What resistance can be supported by a power of 
60 lbs., acting at the extremity of a lever 15 inches long? 

Ans. 16,964 lbs. 

3. The distance from the axis of the trunions of a 2:1111 

weighing 2,016 lbs. to the elevating screw is 3 feet, and the 

distance of the centre of gravity of the gun from the same 

axis is four inches. If the distance between the threads of 

the screw be § of an inch, and the length of the lever 5 inches, 

what power must be applied to sustain the gun in a horizon. 

tal position? Ans. 4.754 lbs. 

The Wedge. 

IOO. The wedge is a solid, bounded by a rectangle 
BD, called the hack y two equal rect- 
angles, AF and DF> called faces ; * 
and two equal isosceles triangles, called 
ends. The line FF, in which the n a , 
faces meet, is called the edge. 

The power is applied at the back, 
to which its direction should be 
norma], and the resistance is applied 
to the faces, and in directions normal 
to them. One half of the resistance 




128 MECHANICS. 

is applied normally to one face, and the other half normally 
to the other face.- Let ABC be a 
section of a wedge made by a plane 
at right angles to the edge. Denote 
the power by JP, and the resistance 
opposed to each face by \B\ denote 
the angle BAC of the wedge by 
2p. Produce the directions of the 
resistances till they intersect in 0. 
This point will be on the line of direc- 
tion of the power. Lay off OF to 
represent the power, and complete 
the parallelogram EB\ then will OF and OB repre- 
sent the resistances developed by the power. Let each 
of the forces \B be resolved into two components, one per- 
pendicular to OF, and the other coinciding with it. The 
two former will be equal and directly opposed to each other, 
whilst the two latter will hold the force P in equilibrium. 
Since BF is perpendicular to FO, and BO perpen- 
dicular to CA, the angle OBF is equal to the angle 
OAC, or <p. The component of ±B in the direction of 
OF, is -Ji?sin;p ; hence, twice this, or Bsrny = P. But 

sins = -XT-.- = ^-, in which b denotes the breadth of the 

CA I ' 

back B C, and I the length of the face CA. Substituting 
this expression for sin<p, and reducing, we have, 

B X±b = PI, or P : B : : ±b : I . (49.) 

That is, the power is to the resistance as one-half of the 
breadth of the back is to the length of the face of the ivedf/e. 

The mechanical advantage of the wedge may be increased 
by diminishing the breadth of the back, or, in other words, 
by making the edge sharper. The principle of the wedge 
finds an important application in all cutting instruments, as 
knives, razors, and the like. By diminishing the thickness 
of the back, the instrument is rendered liable to break, 
hence the necessity of forming cutting instruments of the 
hardest and most tenacious materials. 



ELEMENTARY MACHINES. 129 

General remarks on Elementary Machines. 
101. We have thus far supposed the power and resist- 
ance to be hi equilibrium, through the intervention of the 
machine, their points of application being at rest. If we 
now suppose the point of application to be moved through 
any distance, by the action of an extraneous force, the point 
of application of the power will move through a correspond- 
ing space. These spaces will be described in conformity 
with the design of the machine ; and it will be fouud, in 
each instance, that they are inversely proportional to the 
forces. If we suppose these spaces to be infinitely small, 
they may, in all cases, be regarded as straight lines, which 
will also be the virtual velocities of the forces. If the point 
of application moves in a direction contrary to the direction 
of the resistance, the point of application of the power will 
move in the direction of the power. If we denote the paths 
described by those points respectively, by <5r, and 8p, we 
shall have, 

PSp -. JiSr = 0; or ESp = JZSr . . (50.) 

That is, the algebraic sum of the virtual moments is equal 
to 0. Or, we might enunciate the principle in another man- 
ner, by saying, that in all cases, the quantity of work of the 
power is equal to the quantity of work of the resistance. 

We shall illustrate this principle, by considering a single 
case, that of the single movable pulley, leaving its further 
application to the remaining machines, as exercises for the 
student. 

In the figure, suppose that an extraneous s c x 
force acts to raise the resistance J?, through 
the infinitely small space DE, denoted by Sr ; 
the point of application of P must be raised 



through the infinitely small space FG, denoted aC-^-j 
by 8p, in order that the equilibrium may be li 

preserved. 



E 
In order that the resistance may be raised W 

through the distance DE, both branches of the 

rope enveloping the pulley must be shortened Flg ' w * 

by the same amount ; or, what is the same 

A* 



130 MECHANICS. 

thing, the free end of the rope must ascend through twice 
the distance DM Hence, 

dp — 2Sr. 
But, from the conditions of equilibrium, 

P = \B. 
Multiplying these equations, member by member, we have, 
PSp = BSr. 

Hence, the principle is proved for this particular case. In 
like manner, it may be shown to hold good for all of the 
elementary machines. 

The principle of equality of work of the power and resist- 
ance being true for any infinitely short time, it must neces- 
sarily hold good for any time whatever. Hence, we con- 
clude, that the quantity of work of the power, in overcoming 
any resistance, is equal to quantity of work of the resist- 
ance. Although, by the application of a very small power, 
we are able to overcome a very great resistance, the space 
passed over by the point of application of the power must 
be as much greater than that passed over by the point of 
application of the resistance, as the resistance is greater 
than the power. This is generally expressed by saying, 
that what is gained in power is lost in velocity. 

We see, therefore, that no power is, or can be, gained ; 
the only function of a machine being to enable a smaller 
force to accomplish in a longer time, what a larger force 
would be required to perform in a shorter time. 

Friction. 
102. Friction is the resistance which one body experi- 
ences in moving upon another, the two being pressed 
together by some force. This resistance arises from 
inequalities in the two surfaces, the projections of one sur- 
face sinking into the depressions of the other. In order to 
overcome this resistance, a sufficient force must be applied 



HURTFUL RESISTANCES. 13l 

to break off, or bend down, the projecting points, or else to 
lift the moving body clear of the inequalities. The force 
thus applied, is equal, and directly opposed to the force of 
friction, which is tangential to the two surfaces. The force 
which presses the surfaces together, is normal to them both 
at the point of contact. 

Friction is distinguished as sliding and rolling. The for- 
mer arises when one body is drawn upon another ; the lat- 
ter when one body is rolled upon another. In the case of 
rolling friction, the motion is such as to lift the projecting 
points out of the depressions ; the resistance is, therefore, 
much less than in sliding friction. 

Between certain bodies, the friction is somewhat different 
when motion is just beginning, from what it is when motion 
has been established. The friction developed when a body 
is passing from a state of rest to a state of motion, is called 
friction of quiescence ; that which exists between bodies in 
motion, is called friction of motion. 

The following lavis of friction have been established by 
numerous experiments, viz. : 

First, the friction of quiescence between the same bodies, 
is proportional to the normal pressure, and independent of 
the extent of the surfaces in contact. 

Secondly, the friction of motion between the same bodies, 
is proportional to the normal pressure, and independent, 
both of the extent of the surfaces in contact, and of the 
velocity of the moving body. 

Thirdly, for compressible bodies, the friction of quiescence 
is greater than the friction of motion ; for bodies which 
are sensibly incompressible, the difference is scarcely appre- 
ciable. 

Fourthly, friction may be greatly diminished, by inter- 
posing unguents between the rubbing surfaces. 

Unguents serve to fill up the cavities of surfaces, and thus 
to diminish the resistances arising from their roughness. 
For slow motions and great pressures, the more consistent 
unguents are used, as lard, tallow, and various mixtures • 



132 MECHANICS. 

for rapid motions, and light pressures, oils are generally em- 
ployed. 

The ratio obtained by dividing the entire force of friction 
by the normal pressure, is called the coefficient of friction ; 
the value of the coefficient of friction for any two substances, 
may be determined experimentally as follows : 

Let AJ3 be a horizontal plane 
formed of one of the substances, 
2^\ and let be a cubical block of 



P 



^^ the other substance resting 

upon it. Attach a string OC, 
Ap to the block, so that its direc- 
Fig. 92. tion shall pass through its cen- 

tre of gravity, and be parallel 
to A3 ; let the string pass over a fixed pulley C, and let a 
weight F, be attached to its extremity. 

Increase the weight F till the body just begins to 
slide along the plane, then will this weight measure the 
whole force of friction. Denote this weight by F, that of 
the body, or the normal pressure, by P, and the coefficient 
of friction, by/. Then, from the definition, we shall have, 

f=-' 
J p 

In this manner, values for f corresponding to different 
substances, may be found, and arranged in tables. This 
experiment gives the friction of quiescence. If the weight 
F is such as to keep the body in uniform motion, the 
resulting value of f will correspond to friction of motion. 

The value of/, for any substance, is called the unit, or 
coefficient of friction. Hence, we may define the unit, or 
coefficient of friction, to be the friction due to a normal 
pressure of one pound. 

Having given the normal pressure in pounds, and the 
unit of friction, the entire friction will be found bv multi 
plying these quantities together. 



HURTFUL RESISTANCES. 



133 




Fis. 93. 



There is a second method of finding the value of f ex- 
perimentally, as follows : 

Let AB be an inclined plane, formed of one of the sub- 
stances, and a cubical block, 
formed of the other substance, 
and resting upon it. Elevate the 
plane till the block just begins to 
slide down the plane by its own 
weight. Denote the angle of in- 
clination, at this instant, by a, and 
the weight of 0, by W. Resolve 

the force W into two components, one normal to the sur 
face of the plane, and the other one parallel to it. Denote 
the former component by P, and the latter by Q. Since 
W is perpendicular to A C, and OP to AB, the angle 
WOP is equal to a. Hence, 

P ~ TTcosa, and Q = TFsina. 

The normal pressure being equal to TFcosa, and the force 
of friction being TFsina, we shall have, from the principles 
already explained, 

TTsina BO 



TPcosa 



= tana — 



AG 



The angle a is called the angle of friction. 

Limiting Angle of Resistance. 
103. Let AB be any plane surface, and a body rest- 
ing upon it. Let It be the resultant 
of all the forces acting upon it, in- 
cluding the weight applied at the 
centre of gravity. Denote the angle 
between B and the normal to AB, 
by a, and suppose B to be resolved 
into two components P and §, the 
former parallel to AB, and the latter perpendicular to it ; 
vre shar have, 

P = itsina, and Q — Bcosx. 




134 MECHANICS. 

The friction due to the normal pressure will be equal to 
yifcosx. Now, when the tangential component isteina is 
less than j^cosa, the body will remain at rest ; when it is 
greater than fJlcosx, the body will slide along the plane; 
and when the two are equal, the body will be in a state 
bordering on motion along the plane. Placing the two 
equal, we have, 

/jftcoso, = iisina ; .'. ^f = tanx. 

The value of a is called the limiting angle of resistance, 
and is equal to the inclination of the 

plane, when the body is about to slide ^ 

down by its own weight. If, now, the ^ ; J& 

line OR be revolved about the normal, it \ i '& 

will generate a conical surface, w 7 ithin / ~\y ~7 

which, if any force whatever, including / o / 

the weight, be applied at the centre of Fig. 95. 

gravity, the body will remain at rest, and 
without which, if a sufficient force be applied, the body will 
slide along the plane. This cone is called the limiting cone 
of resistance. 

The values of/*, or the coefficient of friction, in some of the 
most common cases, as determined by Morin, is appended : 

TABLE. 

Bodies between which friction takes place. Coefficient of friction. 

Iron on oak, .62 

Cast iron on oak, .49 

Oak on oak, fibres parallel, .... .48 

Do., do., greased, ....... .10 

Cast iron on cast iron, .15 

Wrought iron on wrought iron, . . .14 

Brass on iron, .16 

Brass on brass, .20 

Wrought iron on cast iron, . . . .19 

Cast iron on elm, .19 

Soft limestone on the same, .... .64 

Hard limestone on the same, . . .38 



HURTFUL RESISTANCES. 13E 

Bodies between which friction takes place Coeffi:ient of friction. 

Leather belts on wooden pulleys, . .47 

Leather belts on cast iron pulleys, . .28 

Cast iron on cast iron, greased, . . .10 

Pivots or axes of wrought or cast iron, on brass or cast 
iron pillows : 

1st, when constantly supplied with oil, .05 

2nd, when greased from time to time, .08 

3rd, without any application, ... .15 

Rolling Friction. 

104. Rolling friction is the resistance which one body 
offers to another when rolling along its surface, the two 
being pressed together by some force. This resistance, like 
that in sliding friction, arises from the inequalities of the 
two surfaces. The coefficient, or unit, of rolling friction is 
equal to the quotient obtained by dividing the entire force 
of friction by the normal pressure. This coefficient is much 
less than the coefficient of sliding friction. 

The following laws of friction have been established, 
when a cylindrical body or wheel rolls upon a plane : 

First, the coefficient of rolling friction is proportional to 
the normal pressure : 

Secondly, it is inversely proportional to the diameter of 
the cylinder or wheel: 

Thirdly, it increases as the surface of contact and velocity 
increase. 

In many cases there is a combination of both sliding and 
rolling friction in the same machine. Thus, in a car upon a 
railroad-track, the friction at the axle is sliding, and that 
between the circumference of the wheel and the track is 
rolling. 

Adhesion. 

105. Adhesion is the resistance which one body ex- 
periences in moving upon another in consequence of the 
cohesion existing between the molecules of the surfaces in 
contact. This resistance increases when the surfaces are 



136 



MECHANICS. 



allowed to remain for some time in contact, and is very 
slight when motion has been established. Both theory and 
experiment show that adhesion between the same surfaces ie 
proportional to the extent of the surface of contact. 

The coefficient of adhesion is the quotient obtained by 
dividing the entire adhesion by the area of the surface of 
contact. Or, denoting the entire adhesion by A, the area 
of the surface of contact by JS, and the coefficient of adhesion 
by a, we have, 

A 



a = 



& 



or A = aS. 



To find the entire adhesion, we multiply the unit ot 
adhesion by the area of the surface of contact. 



Stiffness of Cords. 

106. Let represent a pulley, with a cord AB, 
wrapped around its circumference, and 
suppose a force P, applied at B, to over- 
come the resistance B, and impart motion 
to the pulley. As the rope winds upon the 
pulley, at (7, its rigidity acts to increase the 
arm of lever of B, and to overcome this 
resistance to flexure an additional force is 
required. For the same pulley, this addi- 
tional force may be represented by the 
algebraic expression, 

a + bB, 




Fig. 96. 



in which a and b are constants dependent upon the nature 
and construction of the rope, and B is the resistance to 
be overcome, or the tension of the cord A C. The values of 
a and b for different ropes have been ascertained by experi- 
ment, and tabulated. Finally, if the same rope be wound 
upon pulleys of different diameters, the additional force is 
found to vary inversely as their diameters. If the diameter 
of the pulley be denoted by _Z), and the resistance due to 
stiffness of cordage be denoted by £, we shall have, 



HURTFUL RESISTANCES. 137 

a a + bR 

In the case of the pulley, if we neglect friction, we shall 
have, when the motion is uniform, 

* = *+•+** 

for the algebraic expression of the conditions -of equilibrium. 
The values of a and b have been determined experi- 
mentally for all values of R and D, and tabulated. 

Atmospheric Resistance. 

107. The atmosphere exercises a powerful resistance to 
the motion of bodies passing through it. This resistance is 
due to the inertia of the particles of air, which must be 
overcome by the force of a moving body. It is evident, in 
the first place, other things being equal, that the resistance 
will depend upon the amount of surface of the moving body 
which is exposed to the air in the direction of the motion. 
In the second place, the resistance must increase with the 
square of the velocity of the moving body ; for, if we sup- 
pose the velocity to be doubled, there will be twice as many 
particles met with in a second, and each particle will collide 
against the moving body with twice the force, hence ; if the 
velocity be doubled, the resistance will be quadrupled. By 
a similar course of reasoning, it may be shown that, if the 
velocity be tripled, the retardation will become nine times as 
great, and so on. If, therefore, the retardation correspond- 
ing to a square foot of surface, at any given velocity, be 
determined, the retardation corresponding to any surface 
and any velocity whatever may be computed. 

Influence of Friction on the Inclined Plane. 

108. Let it be required to determine the relation 
between the power and resistance, when the power is just 
on the point of imparting motion to a body up an inclined 
plane, friction being taken into account. 




138 MECHANICS. 

Let AB represent the plane, the body, OP the powe* 
on the point of imparting motion 
up the plane, and OR the weight 
of the body. Denote the power 
by P, the weight by P, the in- 
clination of the plane by a, and 
the angle between the direction 
of the power and the normal to 
the plane by (3. Let P and R 
be resolved into components re- Fi 97 

spectively parallel and perpendi- 

dicular to the plane. We shall have, for the parallel com- 
ponents, Rsina and Psin/3, and for the perpendicular com- 
ponents, Pcosa and Pcos/3. The resultant of the normal 
components will be equal to Rgosol — Pcos/3 ; and, if we 
denote the coefficient of friction by /*, we shall have for the 
entire force of friction (Art. 102), 

/(Rcosx — Pcos/3). 

When we consider the body on the eve of motion up the 
plane, the component Psin/3 must be equal and directly 
opposed to the resultant of the force of friction and the 
component Rsina ; hence, we must have, 

Psin/3 = Rsina, + f (Rcosx — Pcos^). 

Performing the multiplications indicated, and reducing, 
we have, 

P = jB i!E|+^| . . . (51 .) 

( sm/3 -f- fcos3 ) ' 

If we suppose an equilibrium to exist, the body being on 
on the eve of motion down the plane, we shall have. 

Psin/3 -f f (Rgosol — Pcos/3) =z Psina. 

Whence, by reduction, 

P = E \^f^l\ . . . (62 .> 

( sin 3 — /cos/3 \ v 



HURTFUL RESISTANCES. 139 

From these expressions, two values of P may be 
found, when a, /3, /, and B are given. It is evident that 
any value of P greater than the first will „<*use the body to 
slide up the plane, that any value less than the second Till 
permit it to slide down the plane, and that for any inter- 
mediate value the body will remain at rest on the plane. 

If we suppose P to be parallel to the plane, we shall have 
sin/3 = 1, cos/3 = 0, and the two values of P reduce to 

P - B(sina + /cosa) . . . ( 53.) 
and, 

P = i?(sina — /cosa) . . . ( 54.) 

If friction be neglected, we have f = 0; whence, by 

substitution, 

t, j?- P BC 

P = iJsina, or - S =j^i 

a result which agrees with that deduced in a preceding 
article. 

To find the quantity of work of the power whilst drawing 
a body up the entire length of the inclined plane, it may 
be observed that the value of P, in Equation (53), is equal 
to that required to maintain the body in uniform motion 
after motion has commenced. 

Multiplying both members of that equation by AB, we 
have, 

P x AB = B x ^4J5 sina + fjR x A#cosa 

= B x BC +fB x AC. 

But B x BC is the quantity of work necessary to raise 
the body through the vertical height B C ; and fB x AC, 
is the quantity of work necessary to draw the body horizon- 
tally through the distance A C (Art. 75). Hence, the quan- 
tity of work required to draw a body up an inclined plane, 
when the power is parallel to the plane, is equal to the quan- 
tity of work necessary to draw it horizontally across the 
base of the plane, plus the quantity of work necessary to 
raise it vertically through the height of the plane. 



14:0 MECHANICS. 

A curve situated in a vertical plane may be regarded as 
made up of an infinite number of inclined planes. We 
infer, therefore, that the quantity of work necessary to draw 
a body up a curve, the power acting always parallel to the 
direction of the curve, is equal to the quantity of work ne- 
cessary to draw the body over the horizontal projection of 
the curve, plus the quantity of work necessary to raise the 
body through a height equal to the difference of altitude of 
the two extremities of the curve. 

The last two principles enable us to compare the quanti- 
ties of work necessary to draw a train of cars over a hori- 
zontal track, and up an inclined track, or a succession of 
inclined tracks. We may, therefore, compute the length of 
a horizontal track which will consume the same amount of 
work, furnished by the motor, as is actually consumed in 
consequence of the undulation of the track. 

We are thus enabled to compare the relative advantages 
of different proposed routes of railroad, with respect to the 
motive power required for working them. 

Line of Least Traction. 

109. The force employed to draw a body with uniform 
motion along an inclined plane, is called the force of trac- 
tion ; and the line of direction of this force is the line of 
traction. In Equation (51), P represents the force of trac- 
tion required to keep a body in uniform motion up an 
inclined plane, and /3 is the angle which the line of traction 
makes with the normal. It is plain, that when (3 varies, other 
things being the same, the value of P will vary ; there will 
evidently be some value of /3, which will render P the least 
possible ; the direction of P in this case, is called the line of 
least traction ; and it is along this line that a force can be 
applied with greatest advantage, to draw a body up an 
inclined plane. If we examine the expression for P, in 
Equation (51), we see that the numerator remains constant; 
therefore, the expression for P will be least possible when 
the denominator is the greatest possible. By a simple pro* 



HURTFUL RESISTANCES. 141 

cess of the Differential Calculus, it may be shown that the 
denominator -wall be the greatest possible, or a maximum, 
When, 

/ = cot (3, or / = tan(90° — /3). 

That is, the power will be applied most advantageously, 
when it makes an angle with the inclined plane equal to the 
angle of friction. 

From the second value of jP, it may be shown, in like 
manner, that a force will be most advantageously applied, to 
prevent a body from sliding down the plane, when its direc- 
tion makes an angle with the plane equal to the supplement 
of the angle of friction, the angle being estimated as before 
from that part of the plane lying above the body. 

Friction on an Axle. 

110. Let it be required to determine the position of 
equilibrium of a horizontal axle, resting in 
a cylindrical box, when the power is just 
on the point of overcoming the friction 
between the axle and box. 

Let 0' be the centre of a cross section 
of the axle, the centre of the cross sec- 
tion of the box, and JV their point of con- 
tact, when the power is on the point of 
overcoming the friction between the axle 
and box. The element through JSf will be the line of con- 
tact of the axle and box. 

When the axle is only acted upon by its own weight, the 
element of contact will be the lowest element of the box. 
If, now, a power be applied to turn the axle in the direction 
indicated by the arrow-head, the axle will roll up the inside 
of the box until the resultant of all the forces acting upon 
it becomes normal to the surface of the axle at some point 
of the element through JST. This normal force pressing the 
axle against the box, will give rise to a force of friction act- 
ing tangentially upon the axle, which will be exactly equal 
to the tangential force applied at the circumference of the 




142 



MECHANICS. 



axle to produce rotation. If the axle be rolled further up 
the side of the box, it will slide back to JV; if it be moved 
down the box, it will roll back to JV, under the action of the 
force. In this position of the axle, it is in the condition of 
a body resting upon an inclined plane, just on the point of 
sliding down the plane, but restrained by the force of fric- 
tion. Hence, if a plane be passed tangent to the surface of 
the box, along the element iV, it will make with the 
horizon an angle equal to the angle of friction. The rela- 
tion between the power and resistance may then be fc und, 
as in Art. 108. 



BECTILINEAR MOTION. 143 



CHAPTEK V. 

RECTILINEAR AND PERIODIC MOTION. 

Motion. 

111. A material point is in motion when it continually 
changes its position in space. When the path of the moving 
point is a straight line, the motion is rectilinear ; when it is 
a curved line, the motion is curvilinear. When the motion 
is curvilinear, we may regard the path as made up of infi- 
nitely short straight lines ; that is, we may consider it as a 
polygon, whose sides are infinitely small. If any side of this 
polygon be prolonged in the direction of the motion, it will 
be a tangent to the curve. Hence, we say, that a point 
always moves in the direction of a tangent to its path. 

Uniform Motion. 

112. Uniform motion is that in which the moving 
point describes equal spaces in any arbitrary equal portions 
of time. If we denote the space described in one second 
by v, and the space described in t seconds by s, we sha)l 
have, from the definition, 

s = vt ; .'. v = - . . . (55.) 
t 

From the first of these equations, we see that the space 
described in any time is equal to the product of velocity 
and the time / and, from the second, we see that the velo- 
city is equal to the space described in any time, divided by 
t/iat time. 

These laws hold true for all cases of uniform motion. If 
we denote by ds the space described in the infinitely short 
time dt, we shall have, from the last principle, 

• = s (56 -> 



144 MECHANICS. . 

which is the differential equation of uniform motion, v being 
constant. Clearing this equation of fractions, and integ- 
rating, we have, 

s = vt+ C . . . . ( 57.) 

which is the most general equation of uniform motion. If, 
in (57), we make t = 0, we shall have, 

s = a 

Hence, we see that the constant of integration represents 
the space passed over by the point, from the origin of spaces 
up to the beginning of the time t. This space is called the 
initial space. Denoting it by s\ we have, 

s = vt+ s' . . . . (58.) 

If s' — 0, the origin of spaces corresponds to the origin 
of times, and we have, 



the same as the first of Equations (55.) 

Varied Motion. 

113. Varied motion is that in which the velocity is 
continually changing. It can only result from the action 
of an incessant force. 

To find the differential equations of varied motion, let us 
denote the velocity at the time £, by v, and the space 
passed over up to that time, by s. In the succeeding instant 
dt, the space described will be ds, and the velocity gener- 
ated will be dv. Now, the space ds, which is described in 
the infinitely small time dt, may be regarded as having been 
described with the uniform velocity v. Hence, from Equa- 
tion (55), we have, 

v = di < 59 -> 

Let us denote the acceleration due to the incessant force 
at the time t, by <p. We have seen (Art. 24), that the meaa- 



RECTILINEAR MOTION. 145 

ure of the acceleration due to a force, is the velocity that it 
can impart in a unit of time, on the hypothesis that it acts 
uniformly during that time. Now, it is plain that, so long 
as the force acts uniformly, the velocity generated will be 
proportional to the time, and, consequently, the measure of 
the acceleration will be, the quotient obtained by dividing 
the velocity generated in any time, by that time. The quan- 
tity <p is, in general, variable ; but it may be regarded as 
constant during the instant dt ; and from what has just been 
said, we shall have, 

>-% ^ 

Differentiating Equation (59), we have, 
7 d*s 

dv =Ht> 

which, being substituted in Equation (60) gives, 

*■-£ <«•> 

Equations (59), (60), and (61) are the differential equa- 
tions required. The acceleration <p , is the measure of the 
force exerted when the mass moved is the unit of mass 
(Art. 24) ; in any other case, it must be multiplied by the 
mass. Denoting the entire moving force applied to the 
mass mhj F, we shall have, 

d*s 
F = my = m-~ . . . . ( 62.) 

This value of F is the measure of the effective moving 
force in the direction of the body's motion. When a body 
moves upon any curve in space, the motion may be regard- 
ed as taking place in the direction of three rectangular axes. 
If we denote the effective components of the moving force 
in the direction of these axes;, by JT, 3" and Z, the spaces 



146 MECHANICS. 

described being denoted by x, y, and z, we shall have, from 

(62), 

d?x d'y <Pz 

Uniformly Varied Motion. 

1 14. Uniformly varied motion is that in which the 
velocity increases or diminishes uniformly. In the former 
case, the motion is accelerated ; in the latter case, it is re- 
tarded. In both cases, the moving force is constant. De- 
noting the acceleration due to this constant force, by f we 
shall have, from Equation (61), 

8 =/ «*> 

Multiplying by dt, and integrating, we have, 

% = A+0 ■;/... («.) 

ds 
or, since — is equal to v, Equation (59), 
at 

v='ft + ... (65.) 

Multiplying both members of (64) by dt, and integrating, 

we have, 

s = ±ftf + Gt + O ... (66.) 

Equations (65) and (66) express the relations between 
the velocity, space, and time, in the most general case of 
uniformly varied motion. These equations involve the two 
constants of integration C and C, which serve to make 
them conform to the different cases that may arise. To de- 
termine the value of these constants, make t = in the 
two equations, and denote the corresponding values of v 
and 5, by v' and s r . We shall have, 

C =v\ 

' <T = 8'. 



RECTILINEAR MOTION. 14 J 

That is, C is equal to the velocity at the beginning of the 
time t, and O is equal to space passed over up to the same 
time. These values of the velocity and space are called, 
respectively, the initial velocity, and the initial space. 
Substituting for G and C these values in (65) and (66), 
they become, 

v = v '+ft ..... (67.) 

s=z V + v't + iff . .. ( 68.) 

From these equations, we see that the velocity at any 
»ime t, is made up of two parts, the initial velocity, and the 
velocity generated during the time t ; we also see, that the 
space is made up of three parts, the initial space, the space 
due to the initial velocity for the time t, and the space due 
to the action of the incessant force during the same time. 

By giving suitable values to v' and s\ Equations (67) and 
(68) may be made to express every phenomenon of varied 
motion. If we suppose both v' and s' equal to 0, the body 
will move from a state of rest at the origin of times, and 
Equations (67) and (68) will become, 

v = ft (69.) 

s = ifi 2 (70.) 

From the first of these equations, we see that, in uniformly 
varied motion, the velocity varies as the time / and, from 
the second one, we see that the space described varies as 
the square of the time. 

If, in Equation (70), we make t = 1, we have, 

* = if I or > / = 2 *- 

That is, when a body moves from a state of rest, under 
the action of a constant force, the acceleration is equal tc 
twice the space passed over in the first second of time. 

If, in the preceding equations, we suppose f to be essen- 
tially positive, the motion will be uniformly accelerated ; if 
we suppose it to be negative, the motion will be uniformly 



148 MECHANICS. 

retarded. In the latter case, Equations (67) and (68) 
become, 

v = v'-ft ,-. (71.) 

s = a' + v't — iff . . . ( 72.) 

Application to Falling Bodies. 

115. The foece of gravity is the force exerted by the 
earth upon all bodies exterior to it, tending to draw them 
towards it. It is found by observation, that this force is 
directed towards the centre of the earth, and that its intensity 
varies inversely, as the square of the distance from the centre. 

Since the centre of the earth is so far distant from the 
surface, the variation in intensity for small elevations above 
the surface will be inappreciable. Hence, we may re- 
gard the force of gravity at any place on the earth's sur- 
face, and for small elevations at that place, as constant, in 
which case, the equations of the preceding article become 
immediately applicable. The force of gravity acts equally 
upon all the particles of a body, and were there no resistance 
oifered, it would impart the same velocity, in the same time, 
to any two bodies whatever. The atmosphere is a cause of 
resistance, tending to retard the motion of all bodies falling 
through it ; and of two bodies of equal mass, it retards that 
one the most, which offers the greatest surface to the direc- 
tion of the motion. In discussing the laws of falling bodies, 
it will, therefore, be found convenient, in the first place, to 
regard them as being situated in vacuum, after which, a 
method will be pointed out, by means of which the veloci- 
ties can be so diminished, that atmospheric resistance may 
be neglected. 

Let us denote the acceleration due to gravity, at any 
point on the earth's surface, by g, and the space fallen 
through in the time t, by h. Then, if the body moves from 
* state of rest at the origin of times, Equations (69) and 
(70) will give, 

v = gt ( 73.) 

h = \gt ( 74.) 



KECTILTNEAR MOTION. 



149 



From these equations, we see that the velocities at two 
different times are proportional to the times, and the spaces 
to the squares of the times. 

It has been found by experiment that the velocity im- 
parted to a body in one second of time by the action of the 
force of gravity in the latitude of New York, is about 32£ 
feet. Making g = 321 ft., and giving to t the successive 
values I s , 2 s , 3 s , &c., in Equations (73) and (74), we shall 
have the results indicated in the following 

TABLE. 



TIME ELAPSED. 


VELOCITIES ACQUIRED. 


SPACES DESCRIBED. 


^e*« 






. SECONDS. 


FEET. 


FEET. 


1 


321 


16 T V 


2 


64i 


64i 


3 


961 


144| 


4 


128f 


2571 


5 


150|- 


402-J 2 


&c. 


&c. 


&c. 



Solving Equation (74) with respect to % we have, 



t = 



(75.) 



That is, the time required for a body to fall through any 
height is equal to the square root of the quotient obtained 
by dividing twice the height in feet by 32^. 

Substituting this value of t in Equation ( 73), we have. 



or v 2 = 2gh\ 



150 MECHANICS. 

whence, by solving with reference to v and h respectively, 

. . v * 

v = y 2gh, and A — — . . ( 76.) 

These equations are of frequent use in dynamical investiga- 
tions. In them the quantity v is called the velocity due to the 
height A, and the quantity A, the height due to the velocity v. 

If we suppose the body to be projected downwards with 
a velocity v', the circumstances of motion will be made 
known by the Equations, 

v — v' + gt, 

hz= v 't + \gt\ 

In these equations we have supposed the origffr of spaces 
to be at the point at which the body is projected down- 
wards. 

Motion of Bodies projected vertically upwards. 

116. Suppose a body to be projected vertically upwards 
from the origin of spaces with a velocity v\ and afterwards 
to be acted upon by the force of gravity. In this case, the 
force of gravity acts to retard the motion. Making in (71) 
and (72), s f = o, f — g, and s = A, they become, 

v = v' — gt ( 11.) 

h-v't-\gV .... (78.) 

In these equations, A is positive when estimated upwards 
from the origin of spaces, and consequently negative, when 
estimated downwards from the same point. 

From Equation (77), we see that the velocity diminishes 
as the time increases. The velocity will be 0, when, 

v' 
v' — gt = 0, or when t = — ■ 

9 

v' 
If t continues to increase beyond the value — , v will 



RECTILINEAR MOTION. 151 

become negative, and the body will retrace its path. Hence, 
the time required for the body to reach its highest elevation. 
is equal to the initial velocity divided by the force of 
gravity. 

Eliminating t from Equations (77) and (78), we have, 

v n — « 2 
h = 1 !L ( 79.) 

Making v = 0, in the last equation, we have, 

v n 

h = — ( 80.) 

2g v ; 

Hence, the greatest height to which the body will ascend, 
is equal to the square of the initial velocity, divided by 
twice the force of gravity. 

This height is that due to the initial velocity (Art. 115). 

v r 

If, in Equation (77), we make t — t\ we find, 

i/ 

v = g? (81.) 

v' 
If, in the same equation, we make t = \- t\ we find, 

if 

v= -gt' ..... (82.) 

Hence, the velocities at equal times before and after 
reaching the highest points, are equal. 

The difference of signs shows that the body is moving in 
opposite directions at the times considered. 

If we substitute these values of v success: vely, in Equa- 
tion (79), we shall, in both cases, find 



2g 

which shows that the points at which the velocities arc 
equal, both in ascending and descending, are equally distant 
from the highest point ; that is, they are coincident. Hence, 



152 MECHANICS. 

if a body be projected vertically upwards, it will ascend to a 
certain point, and then return upon its path, in such a man- 
ner, that the velocities in ascending and descending will be 
equal at the same points. 

EXAMPLES. 

1. Through what distance will a body fall from a state 
of rest in vacuum, in 10 seconds, and through what space will 
it fall during the last second ? Ans. 16081 ft., and 305^ ft. 

2. In what time will a body fall from a state of rest 
through a distance of 1200 feet ? Ans. 8.63 sec. 

3. A body was observed to fall through a height of 
100 feet in the last second. How long was the body falling, 
and through what distance did it descend ? 

solution. 

If we denote the distance by A, and the time by t, we 
shall have, 

h = %gt\ and h - 100 = \g{t — l) 2 ; 

.*. t — 3.6 sec, and h — 208.44 ft. Ans. 

4. A body falls through a height of 300 feet. Through 
what distance does it fall in the last two seconds ? 

The entire time occupied, is 4.32 sec. The distance fallen 
through in 2.32 sec, is 86. 57 ft. Hence, the distance re- 
quired is 300 ft. - 86.57 ft. =: 213.43 ft. Ans. 

5. A body is projected vertically upwards, with a veloci- 
ty of 60 feet. To what height will it rise ? Ans. 55.9 ft. 

6. A body is projected vertically upwards, with a veloci- 
ty of 483 ft. In what time will it rise to a height of 
1610 feet? 

We have, from Equation (78), 

1610 = 483* - 16 T y 2 ; .-. t = 2 T 8 <nf ± tVV 5 
or, t — 26.2 sec, and t — 3.82 sec. 
The smaller value of t gives the time required ; the larger 



RECTILINEAR MOTION. 153 

value of t gives the time occupied in rising to its greatest 
height, and returning to the point which is 1610 feet from 
the starting point. 

1. A body is projected vertically upwards, with a veloci- 
ty of 161 feet, from a point 214f feet above the earth. In 
what time will it reach the surface of the earth, and with 
what velocity will it strike ? 

SOLUTION. 

The body will rise from the starting point 402.9 ft. The 
time of rising will be 5 sec. ; the time of falling from the 
highest point to the earth will be 6.2 sec. Hence, the re- 
quired time is 11.2 sec. The required velocity is 199 ft. 

8. Suppose a body to have fallen through 50 feet, when 
a second begins to fall just 100 feet below it. How far will 
the latter body fall before it is overtaken by the former ? 

Ans. 50 feet 
Restrained Vertical Motion. 

117. We have seen that the entire force exerted in 
moving a body is equal to the acceleration, multiplied by the 
mass (Art. 24). Hence, the acceleration is equal to the 
moving force, divided by the mass. In the case of a falling 
body, the moving force varies directly as the mass moved ; 
and, consequently j the acceleration is independent of the 
mass. If, by any combination, the moving force can be 
diminished whilst the mass remains unchanged, there will be 
a corresponding diminution in the acceleration. This object 
may be obtained by the combination represented in the 
figure. A represents a fixed pulley, mounted 
on a horizontal axis, in such a manner that the /"T\ 
friction shall be as small as possible ; W and 
W are unequal weights, attached to a flexible 
cord passing over the pulley. If we suppose 
the weight IF greater than IP, the former will *_, 
descend and draw the latter up. If the dif- ' [Jw 
ference is very small, the motion will be very Fig. 99, 

slow, and if the instrument is nicely constructed, 
7* 



w 



\w 



154 MECHANICS. 

we may neglect all hurtful resistances as inap- 
preciable. Denote the masses of the weights 
W and IF', by m and m\ and the force of f A\ 
gravity, by g. The weight W is urged down- 
wards by the moving force mg, and this mo- 
tion is resisted by the moving force m'g. 
Hence, the entire moving force is equal to 3VT 
mg — m'g, or, (m — m')g, and the entire mass F - 
moved, is m -+- m\ since the cord joining the 
weights is supposed inextensible. If we denote the acceh 
eration by g\ we shall have, from what was said at the 
beginning of this article, 

m — m' x 

q' = ■ -,q (83.) 

By diminishing the difference between m and m\ we may 
make the acceleration as small as we please. It is plain that 
g' is constant; hence, the motion of IF is uniformly varied. 

tyyt /yyi ' 

If we replace g by — — — - r g, in Equations (73) and (74), 

they will make known the circumstances of motion of the 
body W. This principle is employed to illustrate the laws 
of falling bodies by means of Atwood's machine. 

Had the two weights under consideration been attached 
to the extremities of cords passing around a wheel and its 
axle, and in different directions, it might have been shown 
that the motion would be uniformly varied, when the mo- 
ment of either weight exceeded that of the other. The 
same principle holds good in the more complex combinations 
of pulleys, wheels and axles, &c. In practice, however, the 
hurtful resistances increase so rapidly, that even when the 
moving force remains constant, the velocity soon attains 
a maximum limit, after which the motion will be sensibly 
unifo 1 m. 

EXAMPLES. 

1. Two weights of 5 lbs. and 4 lbs., respectively, are 
suspended from the extremities of a cord passing over a 



RECTILINEAR MOTION. 155 

fixed pulley. What distance will each weight describe in the 
first second of time, what velocity will be generated in one 
second, and what will be the tension of the connecting cord ? 

SOLUTION. 

Since the masses are proportional to the weights, we 
shall have, 

^ = ^ = i x3 4 ft -.= 3 - 5 ' 4ft - 

Hence, the velocity generated is 3.574 ft., and the space 
passed over is 1.787 ft. To find the tension of the string, 
denote it by x. The moving force acting upon the heavier 
body, is (5 — x)g, and the acceleration due to this force, 

( — - — \g\ the moving force acting upon the lighter body, 

/SY> A\ 

is {x— 4)<7, and the corresponding acceleration, ( — - — \g. 

But since the two bodies move together, these accelerations 
must be equal. Hence, 

/5 — x\ fx — 4\ 

(— > = (— > ; 

.*. x = 4|- lbs., the required tension. 

2. A weight of 1 lb., hanging on a pulley, descends and 
drags a second weight of 5 lbs. along a horizontal plane. 
Neglecting hurtful resistances, to what will the accelerating 
force be equal, and through what spruce will the descending 
body move in the first second ? 

SOLUTION. 

The moving force is equal to 1 x g, and the mass moved 
is equal to 6. Hence, the acceleration is equal to - = 5.3622 
ft., and the space described will be equal to 2.6811 ft. 

3. Two bodies, each weighing 5 lbs., are attached to a 
string passing over a fixed pulley. What distance will each 



ao6 



MECHANICS. 



body move in 10 seconds, when a pound weight is added to 
one of them, and what velocity will have been generated at 
the end of that time ? 

SOLUTION. 

The acceleration will be equal to T \g = 2.924 ft. = g'. 
But, s = \g'tf, v = g't. Hence, the space described in 10 
eaconds is 146.2 ft., and the velocity generated is 29.24 ft. 

4. Two weights, of 16 oz. each, are attached to the ends 
of a string passing over a fixed pulley. What weight must 
be added to one of them, that it may descend through a 
foot in two seconds ? 

solution. . 
Denote the required weight by x ; the acceleration will 



be equal to 



g z= g'. But s = ^g'tf : making s = 1 



1 = 



X 32i; 



x = 0.505 oz. Ans. 



32 + x 
and t = 2, we have, 

2x 
32 + x 

Atwood's Machine. 
118. Atwood's machine is a contrivance to illustrate the 
laws of falling bodies. It consists of a vertical 
post AB, about 12 feet in height, supporting, 
at its upper extremity, a fixed pulley A. To 
obviate, as much as possible, the resistance of 
friction, the axle is made to turn upon friction 
rollers. A fine silk string passes over the 
pulley, and at its two extremities are fastened 
two equal weights G and D. In order to 
impart motion to the weights, a small weight 
G, in the form of a bar, is laid upon the 
weight (7, and by diminishing its mass, the 
acceleration may be rendered as small as 
desirable. The vertical rod AI>, graduated 
to feet and decimals, is provided with two 
sliding stages E and F\ the upper one is in 
the form of a ring, which will permit the 




Fig. 100. 



RECTILINEAR MOTION. 157 

weight (7, to pass, but not the bar G ; the lower one is in 
the form of a plate, which is intended to intercept the 
weight G. There is also connected with the instrument a 
seconds pendulum for measuring time. 

Let us suppose that the weights of G and J9, are each 
equal to 181 grains, and that the weight of the bar G, is 
24 grains. Then will the acceleration be 

a' = 2 - g = 2 ft. ; 

u 362 + 24 y ' 

and since h = ig't*, and v = g't (Art. 115), we shall 
have, for the case in question, 

h = f, and v = 2t. 

If, in these equations, we make t = 1 sec, we shall 
have h = 1, and v ==. 2. If we make t = 2 sec, we shall, 
in like manner, have A = 4, and y = 4. If we* make 
£ = 3 sec, we shall have h = 9, and w = 6, and so on. 
To verify these results experimentally, commencing with 
the first. The weight G is drawn up till it comes opposite 
the of the graduated scale, and the bar G is placed upon 
it. The weight thus set is held in its place by a spring. 
The ring E is set at 1 foot from the 0, and the stage JF is 
set at 3 feet from the 0. When the pendulum reaches one 
of its extreme limits, the spring is pressed back, the weight 
(7, G descends, and as the pendulum completes its vibration, 
the bar G strikes the ring, and is retained. The acceleration 
then becomes 0, and the weight G moves on uniformly, with 
the velocity that it had acquired, in the first second ; and it 
will be observed that the weight G strikes the second stage 
just as the pendulum completes its second vibration. Had 
the stage F been set at 5 feet from the 0, the weight G 
would have reached it at the end of the third vibration of 
the pendulum. Had it been 7 feet from the 0, it would 
have reached it at the end of the fourth vibration, and so on. 

To verify the next result, we set the ring E at four feet 



158 MECHANICS. 

from the 0, and the stage F at 8 feet from the 0, and pro- 
ceed as before. The ring will intercept the bar at the end 
of the first vibration, and the weight will strike the stage at 
the end of the second vibration, and so on. 

By making the weight of the bar less tnan 24 grains, the 
acceleration is diminished, and, consequently, the spaces and 
velocities correspondingly diminished. The results may be 
verified as before. 

Motion of Bodies on Inclined Planes. 

119. If a body be placed on an inclined plane, and 
abandoned to the action of its own weight, it will either 
slide or roll down the plane, provided there be no friction 
between it and the plane. If the body is spherical, it will 
roll, and in this case the friction may be disregarded. Let 
the weight of the body be resolved into two components ; 
one perpendicular to the plane, and the other parallel to it. 
The plane of these components will be vertical, and it will 
also be perpendicular to the given plane. The effect of the 
first component will be counteracted by the resistance of the 
plane, whilst the second component will act as a constant 
force, continually urging the body down the plane. The 
force being constant, the body will have a uniformly varied 
motion, and Equations (67) and (68) will be immediately 
applicable. The acceleration will be found by projecting 
the acceleration due to gravity upon the inclined plane. 

Let AP represent a section of the inclined plane made by 
a vertical plane taken perpendicular 
to the given plane, and let P be the 
centre of gravity of a body resting 
on the given plane. Let PQ repre- 
sent the acceleration due to gravity, 
denoted by g, and let PR be the Ficr 101 

component of g, which is parallel to 

AB, denoted by g', PS being the normal component. 
Denote the angle that A JB makes with the horizontal plane 
by a. Then, since PQ is perpendicular to PC, and QP to 




RECTILINEAR MOTION. 159 

AB, the angle RQP is equal to ABC, or to a. Hence we 
have, from the right-angled triangle PQB, 

g' = #sina. 

But the triangle ABC is right-angled, and, if Ave denote 
its height AG by A, and its length AB by I, we shall have 

sina = y , which, being substituted above, gives, 

'• = £..'.' (84.) 

This value of g' is the value of the acceleration due to the 
moving force. Substituting it for f in Equations (67) and 
(68), we have, 

v = v' + ^t, 

• =v.+ trt + §*. 

If the body starts from rest at A, taken as the origin of 
spaces, then will v' = and s' = 0, giving, 

* = l£l (85.) 

•-#/ < 86 -) 

To find the time required for a body to move from the 
top to the bottom of the plane, make s = I, in (86) ; there 
will result, 

'=fh ••• «=v?;i»-> 

Hence £/ie fo'me varies directly as the length, and inversely 
as the square root of the height. 

For two planes having the same height, but different 
lengths, the radical factor of the value of t will remain con- 



160 MECHANICS. 

stant. Hence, tne limes required for a body to move doimi 
any two planes having the same height, are to each other as 
their lengths. 

To determine the velocity with which a body reaches the 
bottom of the plane, substitute for % in Equation (85) its 
value taken from Equation (86). We shall have, after 
reduction, 

v = ■y/Zgh. 

But this is the velocity due to the height h (Art. 115). 
Hence, the velocity generated in a body whilst moving 
down any inclined plane, is equal to that generated in 
falling freely through the height of the plane. 

EXAMPLES. 

1. An inclined plane is 10 feet long and 1 foot high. 
How long will it take for a body to move from the top to 
the bottom, and what velocity will it acquire in the 
descent ? 

SOLUTION. 

We have, from Equation (87), 

IT 



substituting for I its value 10, and for h its value 1, we have, 
t = 2^ seconds nearly. 

From the formula v = -\fiigh, we have, by making 
h = 1, 

V = -v/64.33 = 8.02 ft. 

2. How far will a body descend from rest in 4 seconds, 
on an inclined plane whose length is 400 feet, and whose 
height is 300 feet ? Ans. 193 ft. 

3. How long will it take for a body to descend 100 feet 
on a plane whose length is 150 feet, and whose height is 60 
feet? Ans. 3.9 sec 



RECTILINEAR MOTION. 161 

4. There is an inclined railroad track, 2^ miles in length, 
whose inclination is 1 in 35. What velocity Trill a car 
attain, in running the whole length of the road, by its own 
weight, hurtful resistances being neglected ? 

Ans. 155.75 ft., or, 106.2 m. per hour. 

5. A railway train, having a velocity of 45 miles per 
hour, is detached from the locomotive on an ascending grade 
of 1 in 200. How far, and for what time, will the train 
continue to ascend the inclined plane ? 

SOLUTION. 

We find the velocity to be 66 ft. per second. Hence, 
66 = \/2gh ; or, h = 67.7 ft. for the vertical height. 
Hence, 67.7 X 200 = 13,540 ft., or, 2.5644 m., the distance 
which the train will proceed. We have, 

t = I \J —t = 410.3 sec, or, 6 min. 50.3 sec, 
foi the time required to come to rest. 

6. A body weighing 5 lbs. descends vertically, and draws 
a weight of 6 lbs. up an inclined plane of 45°. How far 
will the first body descend in 10 seconds ? 

SOLUTION. 

The moving force is equal to 5 — 6 sin 45° ; and, conse_ 
sequently, the acceleration, 

5-6sin45° .757 ^ 0010 
f ss — nT — = _ = .068818; 

.-. s = Ig'f = 3.4409 ft. Ans. 

Motion of a Body down a succession of Inclined Planes. 

120. If a body start from the top of an inclined plane, 
with an initial velocity v\ it will reach the bottom with a 
velocity equal to the initial velocity, increased by that due 
to the height of the plane. This velocity, called the terminal 
velocity, will, therefore, be equal to that which the body 




162 MECHANICS. 

would have acquil 3d by falling freely through a height equal 
to that due to the initial velocity, increased by that of the 
plane. Hence, if a body start from 
a state of rest at A, and, after having 
passed over one inclined plane AB, 
enters upon a second plane BC, 
without loss of velocity, it will reach Fi 102 

the bottom of the second plane with 

the same velocity that it would have acquired by falling 
freely through DC, the sum of the heights of the two 
planes. Were there a succession of inclined planes, so ar- 
ranged that there would be no loss of velocity in passing 
from one to another, it might be shown, by a similar course 
of reasoning, that the terminal velocity would be equal to 
that due to the vertical distance of the terminal point below 
the point of starting. 

By a course of reasoning entirely analagous to that em- 
ployed in discussing the laws of motion of bodies projected 
vertically upwards, it might be shown that, if a body were 
projected upwards, in the direction of the lower plane, with 
the terminal velocity, it would ascend along the several 
planes to the top of the highest one, where the velocity 
would be reduced to 0. The body would then, under the 
action of its own weight, retrace its path in such a manner 
that the velocity at every point in descending would be the 
same as in ascending, but in a contrary direction. The time 
occupied by the body in passing over any part of its path in 
descending, would be exactly equal to that occupied hi 
passing over the same portion in ascending. 

In the preceding discussion, we have supposed that there 
is no loss of velocity in passing from one plane to another. 
To ascertain under what circumstances this condition will be 
fulfilled, let us take the two planes AB and B C. Prolong 
BG upwards, and denote the angle ABE, by (p. Denote 
the velocity of the body on reaching B, by v'. Let v' be 
resolved into two components, one in the direction of B C, 
and the other at right angles to it. The effect of the lattei 



PERIODIC MOTION. 163 

will be destroyed by the resistance of the plane, and the 
former will be the eifective velocity in the direction of the 
plane B C. From the rule for decomposition of velocities, 
we have, for the effective component of v\ the value v' cos?. 
Hence, the loss of velocity due to change of direction, is 
t/ _ v' coscp ; or, v'(l — cos?), which is equal to v' ver-sin?. 
But when <p is infinitely small, its versed-sine is 0, and there 
will be no loss of velocity. Hence, the loss of velocity due 
to change of direction will always be 0, when the path of 
the body is a curved line. This principle is general, and 
may be enunciated as follows : When a body is constrained 
to describe a curvilinear path, there will be no loss of velo- 
city in consequence of the change in direction of the body's 
motion. 

Periodic Motion. 

121. Periodic motion is a kind of variable motion, in 
which the spaces described in certain equal periods of time 
are equal. This kind of motion is exemplified in the pheno- 
mena of vibration, of which there are two cases. 

1st. Rectilinear vibration. Theory indicates, and experi- 
ment confirms the fact, that if a particle of an elastic fluid 
be slightly disturbed from its place of rest, and then aban- 
doned, it will be urged back by a force, varying directly as 
its distance from the position of equilibrium ; on reaching 
this position, the particle will, by virtue of its inertia, pass 
to the other side, again to be urged back, and so on. To 
determine the time required for the particle to pass from 
one extreme position to the opposite one and back, let us 
denote the displacement at any time t by s, and the accelera- 
tion due to the restoring force by <p ; then, from the law of 
the force, we shall have 9 = n 2 s, in which n is constant for 
the same fluid at the same temperature. Substituting for 9 
its value, Equation (61), and recollecting that 9 acts in a 
direction contrary to that in which s is estimated, we have, 

-ar= ns 



1 64: MECHANICS. 

Multiplying both members by 2<Zs, we have, 
dt 2 ' 



whence, by integration, 
ds* 



*—*+ +0 ±-* 



The velocity v will be when s is greatest possible; 
denoting this value of s by a, we shall have, 

rtfa? -f- G = 0; whence, C = — ft/V. 

Substituting this value of (7 in the preceding equation, it 
becomes, 

?; a = -=-r = ft/ 2 (a 2 — s 2 ) ; whence, ndt = — - . ( 88.) 

Integrating the last equation, we have, 

nt + O — sin- 1 - . . . . (89.) 
a 

Taking the integral between the limits s — -\- a and 
s = — a, and denoting the corresponding time by \r 
r being the time of a double vibration, we have, 

\nr — ir ; whence, r = — • 
2 n 

The value of t is independent of the extent of the excur- 
sion, and dependent only upon n. Hence, in the same 
medium, and at the same temperature, the time of vibration 
is constant. 

These principles are of utility in discussing the subjects 
of sound, light, &c. 



PERIODIC MOTION. 165 

2ndly. Curvilinear vibration. Let ABG be a vertical 

plane curve, symmetrical with 

respect to DB. Let AG he a 

horizontal line, and denote the 

distance EB by h. If a body /' 

were placed at A and abandoned <£;!'_ 

to the action of its own weight, £n^. 

beinsj constrained to remain on 

. jj 

the curve, it would, in accord- Fi 103 

ance with the principles of the 

last article, move towards B with an accelerated motion, 
and, on arriving at B, would possess a velocity due to the 
height h. By virtue of its inertia, it would ascend the 
branch B G with a retarded motion, and would finally reach 
(7, where its velocity would be 0. The body would then be 
in the same condition that it was at A, and would, conse- 
quently, descend to B and again ascend to A, whence it 
would again descend, and so on. Were there no retarding 
causes, the motion would continue for ever. From what 
has preceded, it follows that the time occupied by the body 
in passing from A to B is equal to that in passing from B 
to (7, and also the time in passing from C to B is equal to 
that in passing from B to A. Further, the velocities of the 
body when at G and IT, any two points lying on the same 
horizontal, are equal, either being that due to the height 
EK. These principles are of utility in discussing the 
pendulum. 

Angular Velocity. 

122. When a body revolves about an axis, its points 
being at different distances from the axis, will have different 
velocities. The angular velocity is the velocity of a point 
whose distance from the axis is equal to 1. To obtain the 
velocity of any other point, we multiply its distance from 
the axis by the angular velocity. To find a general expres- 
sion for the velocity of any point of a revolving body, let us 
denote the angular velocity by w, the space passed over by 
a point at the unit's distance from the axis m the time dt y 



166 



MECHANICS. 



by dd. The quantity db is an infinitely small arc, having a 
radius equal to 1 ; and, as in Art. 113, it is plain that we 
may regard the angular motion as uniform, during the infin- 
itely small time dt. Hence, as in Article 113, we have, 



dt 



(90.) 



If we denote the distance of any point from the axis by I, 
and its velocity by w, we shall have, 



lu ; or, v = I 



dt 



(91.) 



The Simple Pendulum. 

123. A pendulum is a heavy body suspended from a 
horizontal axis, about which it is free to vibrate. In order 
to investigate the circumstances of vibration, let us first 
consider the hypothetical case of a single material point 
vibrating about an axis, to which it is attached by a rod des- 
titute of weight. Such a pendulum is called a simple pen- 
dulum. The laws of vibration, in this case, will be identical 
with those explained in Art. 121, the arc ABC being the 
arc of a circle. The motion is, therefore, periodic. 

Let ABC be the arc through 
which the vibration takes place, and 
denote its radius by I. The angle 
CD A is called the amplitude of vi- 
bration; half of this angle ABB, 
denoted by a, is called the angle of 
deviation ; and I is called the length 
of the pendulum. If the point starts 
from rest, at A, it will, on reaching 
any point H, of its path, have a velocity v, due to the height 
EK, denoted by h. Hence, 

v = V%gh (92. 

If we denote the variable angle IIDB by 6, we shall 




PERIODIC MOTION. 167 

have DK = Zcos4 ; we shall also have DE = fcosa ; and 
since h is equal to DK — DJE^ we shall have, 

h — I (cosd — cosa). 

Which, being substituted in the preceding formula, gives, 

v — \/2gl(cos) — cosa). 
From the preceding article, we have, 



dt 



Equating these two values of v, we have, 



l-T.= \/2gl(cosd — cosa). 
ut 

Whence, by solving with respect to dt, 

dt = k/-- , , . . (93.) 

V 2g y cos;) — cosa ' 

If we develop cos# and cosa into series, by McLaukin's 
theorem, we shall have, 

cosd = 1 1 — &c. ; 

2 1.2.3.4 ' 



COSa =1 — -f- — — - - &G. 

2 2.3.4 

When a is very small, say one or two degrees, 6 being 
still smaller, we may neglect all the terms after the second 
as inappreciable, giving 



= 1 -P 



a« 
cosa =1 — ; 

2 ' 



168 MECHANICS. 

or, cos0 — cosa — i(a 3 _ &'), 

Substituting in Equation (93), it becomes, 



*=v^vra • • • • m -> 

Integrating Equation (94), we have, 

V # a 

Taking the integral between the limits 6 = — a, and 
d = + «,, £ will denote the time of one vibration, and we 
shall have, 

t = * iA (95.) 

Hence, the time of vibration of a simple pendulum is 
equal to the number 3.1416, multiplied into the square root 
of the quotient obtained by dividing the length of the pen- 
dulum by the force of gravity. 

For a pendulum, whose length is l\ we shall have, 

From Equations (95) and (96), we have, by division, 

j=^fn or, f.t':: yT: Vf ( 97.) 

That is, the times of vibration of two simple pendulums, 
are to each other as the square roots, of their lengths. 

If we suppose the lengths of two pendulums to be the 
same, but the force of gravity to vary, as it does slightly in 
different latitudes, and at different elevations, we shall have, 

t = * \/-, and t" ~ ^ \ —. « 
\ g" V g" 



PERIODIC MOTION. 169 

Whence, by division, 



* 9 



t" ~ V g ' 



or. 



*:*":: V7 7 " : VF • ( »8.) 



That is, £/*e tames of vibration of^ the same simple pen- 
dulum, at two different places, are to each other inversely as 
the square roots of the forces of gravity at the two places. 

If we suppose the times of vibration to be the same, and 
the force of gravity to vary, the lengths will vary also, and 
vre shall have, 

-, and t = if\/—. 
Equating these values and squaring, we have, 

-= l -,\ or, (:*:.:,:/.. . (99.) 

O if 

That is, the lengths of simple pendulums which vibrate in 
equal times at different places, are to each other as the 
forces of gravity at those places. 

Vibrations of equal duration are called isochronal. 

The Compound Pendulum. 

124. A compound pendulum is a heavy body free to 
oscillate about a horizontal axis. This axis is called the 
axis of suspension. The straight line drawn from the 
centre of gravity of the pendulum perpendicular to the axis 
of suspension is called the axis of the pendulum. 

In all practical applications, the pendulum is so taken that 
the plane through the axis of suspension and the centre of 
gravity divides it symmetrically. 

Were the elementary particles of the pendulum entirely 
disconnected, but constrained to remain at invariable dis- 
tances from the axis of suspension, we should have a col- 
lection of simple pendulums. Those at equal distances from 
8 



170 MKCHANICS. 

the axis would vibrate in equal times; those unequally 
distant from it would vibrate in unequal times. 

Those particles which are at the same distance from the 
axis of suspension lie upon the surface of a cylinder, whose 
axis coincides with the axis of suspension, and we may, with- 
out at all affecting the time of vibration, suppose them all to 
be concentrated at the point in which the cylinder cuts the 
axis of the pendulum. If we suppose the same to be done for 
each of the concentric cylinders, we may regard the pendu- 
lum as made up of a succession of heavy points, a, b, . . . p, k, 
lying on the axis, firmly connected with each 
other and with the point of suspension C. 
The particles a, b, &c, nearest to C will tend 
to accelerate the motion of the entire pendu- , 

lum, whilst those most remote, as p, k, &c, / 

will tend to retard it. There must, therefore, / 

be some intermediate point, as 0, which will /--■ 

vibrate precisely as though it were not con- Ti<r 105 

nected with the system ; were the entire mass 
of the pendulum concentrated at this point it would vibrate 
in the same time as the given pendulum. This point is 
called the centre of oscillation. Hence, the centre of oscil- 
lation of a pendulum is that point of its axis, at which, if 
the entire mass of the pendulum were concentrated, its time 
of vibration would be unchanged. A line drawn through 
this point, parallel to the axis of suspension, is called the axis 
of oscillation. The distance from the axis of oscillation to 
the axis of suspension is the length of an equivalent simple 
pendulum, that is, of a simple pendulum, whose time of 
vibration is the same as that of the compound pendulum. 

To find an expression for CO, C being the axis of sus- 
pension, and the axis of oscillation. Denote CO by I; 
let G be the centre of gravity, and denote the distance 
CG by k; denote the masses concentrated at a, b, . . . p, k, 
by m, m' . . . m", m'", and their distances from C by 
r, r' . . . r'\ r"'. 

Whatever may be the position of CO, the effective com 



PERIODIC MOTION. 171 

ponent of gravity is the same for each particle, and were 
they free to move, each would have impressed upon it the 
same velocity that is actually impressed upon Denote 
the angular velocity at any instant, by w ; then will the 
actual velocity of the mass m, be equal to rw, and the effec- 
tive moving force will be equal to mrw (Art. 24). Had the 
mass m been at (9, instead of at a, the entire moving force 
impressed would have been effective, and its measure would 
have been mlu. The difference between these forces, or 
m(l — r)u, is that portion of the force applied at a which 
goes to accelerate the motion of the system. The moment 
of this force with respect to C, is m(l — r)ru. In like 
manner, for the force acting at #, which also tends to accel- 
erate the system, we have m r (l — r')r'w, and so on, for all 
of th e particles between and C. By a similar course of 
reasoning, we get, for the moments of the force tending to 
retard the system, and which are applied at the points 
p, &, <fcc, m"{r" - l)r"u, m"\r'" — l)r'"w, &c. But since 
there is neither acceleration nor retardation, in consequence 
of the action of these forces, they must be in equilibrium, 
and, consequently, the sum of the moments of the forces 
which tend to accelerate the system, must be equal to the 
sum of the moments, which tend to retard the system 
Hence, we have, 

m(l — r)ru + m'(l — r')r'w -f &c. 

= m "{r" - iy u + m »Y" - l)r'"u + &c. 

Striking out the factor w, and reducing, we have, 
(mr + «jV'+ m"r" + &c.) I = mr 2 + m'r n -f m' 'r" 2 + &c, 
or, 2(mr) X I = 2(w 2 ). 

Hence, 

1=5^ . . . . (100.) 



172 



MECHANICS. 



The expression 2(mr ! ), is called the moment of inertia 
of the body with respect to the axis of suspension. 

The moment of inertia of a body, with respect to any 
axis, is the algebraic sum of the products obtained by mul- 
tiplying the mass of each elementary particle by the square 
of its distance from the axis. 

The expression 2(mr), is called the moment of the mass, 
with respect to the axis of suspension. 

The moment of the mass with respect to any axis, is the 
algebraic sum of the products obtained by multiplying the 
mass of each elementary particle by its distance from the 
axis. 

From the principle of moments, this is equal to the mo- 
ment of the entire mass, concentrated at the centre of grav- 
ity. Denote the mass, or 2(m), by M, the distance of its 
centre of gravity from the axis, by k, and we shall have, 

2(mr) = M k ......( 101.) 

Substituting this in Equation (100), we have, 

I- Mk ...... (102.) 

That is, the distance from the axis of suspension to the 
axis of oscillation is equal to the moment of inertia, taken 
with respect to the axis of suspension, divided by the moment 
of the mass, taken with respect to the same axis. 

Let the axis of oscillation be taken as an axis of suspen- 
sion, and denote its distance from the new axis of oscillation 
by V. The distances of a, b . . . p, k, from 0, will be 
I — r, I — r', &c, and the distance G will be I — k 
From the principle just enunciated, we shall have, 

_ 2[m(l-ry] 
~ M(l-k) 



PERIODIC MOTION. 173 

Or, performing the operation of squaring and reducing, 

_ 2(raZ 2 - 2mrl + rar 2 ) _ 2(mF) — 22(mrl) + S(mr ') 
~ M{1- k) ~ ~~M\l - k) 

But I is constant, hence 2(mZ 2 ) = 2(m) x ? 2 = MP, 
also, 22{mrl) = 22 (mr) X J = 2JfAtf; from Equation (102) 
we have, 2(mr 2 ) = JfM Substituting these values in the 
preceding equation, we have, 

_ Ml 3 - 2Mkl + iltf kl __ itf(£ - &)£ 
M(l-k) ~~ M{1 - k) ' 

or, 



(103.) 



Henee, it follows that the axes of suspension and oscilla- 
tion are convertible ; that is, if either be taken as the axis 
of suspension, the other will be the axis of oscillation, and 
the reverse. 

This property of the compound pendulum has been em- 
ployed to determine experimentally the length of the 
seconds pendulum, and the value of the force of gravity at 
different places on the surface of the earth. 

A straight bar of iron CD, is provided with two knife- 
edge axes, A and B, of hardened steel, at right 
angles to the axis of the bar, and having their 
edges turned towards each other. These axes are 
so placed that their plane will pass through the 
axis of the bar. The pendulum thus constructed is 
suspended on horixontal plates of polished agate, 
and allowed to vibrate about each axis in turn till, 
by filing away one of the ends of the bar, the times 
of vibration about the two axes are made equal. W 
The distance AB is then equal to the length of the F . 106 
equivalent simple pendulum; that is, of a simple 
pendulum which will vibrate in the same time as the bai 
about either axis. 



174 MECHANICS. 

To employ the pendulum thus adjusted to find the length 
of a simple seconds pendulum at any place, the pendulum is 
carefully suspended, and allowed to vibrate through a very 
small angle ; the number of vibrations is counted, and the 
time occupied is carefully noted by means of a well-regulated 
chronometer. The entire time divided by the number of 
vibrations performed, gives the time of a single vibration. 
The distance between the axes is carefully measured by an 
accurate scale of equal parts, which gives the length of the 
corresponding simple pendulum. To find the length of the 
simple seconds pendulum, we then make use of Proportion 
(97), substituting in it for t' and V the values just found, and 
for t, 1 second ; the only remaining quantity in the propor- 
tion is £, which may be found by solving the proportion. 
This value of I is the required length of the simple seconds 
pendulum at the place where the observation is made. In 
making the observations, a variety of precautions must be 
taken, and several corrections applied, the explanation of 
which does not fall within the scope of this treatise. It is 
only intended to point out the general method of proceed- 
ing. By a long series of carefully conducted experiments, 
it has been found that the length of a simple seconds pen- 
dulum in the Tower of London is 3.2616 ft., or 39.13921 in. 
By a similar course of proceeding, the length of the seconds 
pendulum has been determined for a great number of places 
on the earth's surface, at different latitudes, and from these 
results the corresponding values of the force of gravity at 
those points have been determined according to the following 
principle : 

Fiom Equation (95), which is, t ==. if\/— , we find, by 

if 

solving with respect to g, and making t = 1, 

g = *n. 

From this equation the value of g may be found at 
different places, by simply substituting for I the length of the 



PERIODIC MOTION. 175 

seconds pendulum at those places. In this manner, the value 
nf g is found for a great number of places in different 
latitudes, and from these values the form of the earth's 
surface may be computed. 

It has been ascertained in this manner that if the force of 
gravity at any point on the earth's surface be denoted by g, 
the force of gravity at a point whose latitude is 45°, by g\ 
and the latitude of the place where the force of gravity is 
g', by Z, we shall have, 

g — g\\ — .002695cos2Z). 



PRACTICAL APPLICATIONS OF THE PENDULUM. 

125. One of the most important of the applications of 
the pendulum is to regulating the motion of clocks. A 
clock consists of a train of wheelwork, the last wheel of the 
train connecting with the upper extremity of a pendulum- 
rod by a piece of mechanism called an escapement. The 
wheelwork is maintained in motion by means of a descending 
weight, or by the elastic force of a coiled spring, and the 
wheels are so arranged that one tooth of the last wheel in 
the train escapes from the upper end of the pendulum-rod 
at each vibration of the pendulum, or at each beat. The 
number of beats is registered and rendered visible on a 
dial-plate by means of indices, called the hands of the clock. 

On account of the expansion and contraction of the ma- 
terial of which the pendulum is composed, the length of the 
pendulum is liable to continual variation, which gives rise to 
an irregularity in the times of vibration of the pendulum. 
To obviate this inconvenience, and to render the times of 
vibration perfectly uniform, several ingenious devices have 
been resorted to, giving rise to what are called compensating 
pendulums. We shall indicate two of the most important 
of these combinations, observing that all of the remaining 
ones are nearly the same in principle, differing only in the 
modes of application. 



176 



MECHANICS. 



Graham's Mercurial Pendulum. 

126. Graham's mercurial pendulum consists of a rod oi 
steel about 42 inches long, branched towards its lower end ; 
so as to embrace a cylindrical glass vessel 7 or 8 inches deep, 
and having 6.8 in. of this depth filled with mercury. The 
exact quantity of mercury being dependent on the weight 
and expansibility of the other parts oi the pendulum, must 
be determined by experiment in each individual case 
When the temperature increases, the steel rod is lengthened, 
and, at the same time, the mercury expanding, rises in the 
cylinder. When the temperature decreases, the steel bar is 
shortened, and the mercury falls in the cylinder. By a 
proper adjustment of the quantity of mercury, the effect 
of the lengthening or shortening of the rod is exactly coun- 
terbalanced by the rising or falling of the centre of gravity 
of the mercury, and the axis of oscillation is kept at an 
invariable distance from the axis of suspension. 



Harrison's Gridiron Pendulum. 

127. Harrison's gridiron pendulum consists of five 
rods of steel and four of brass, placed alter- 
nately with each other, the middle rod, or that 
from which the bob is suspended, being of steel. 
These rods are connected by cross-pieces in 
such a manner that, whilst the expansion of the 
steel rods tends to elongate the pendulum, or 
lower the bob, the expansion of the brass rods 
tends to shorten the pendulum, or raise the bob. 
By duly proportioning the sizes and lengths of 
the bars, the axis of oscillation may be main- 
tained, by the combination, at an invariable dis- 
tance from the axis of suspension. From what 
has preceded, it follows that whenever the dis- 
tance from the axis of oscillation to the axis of s 
remains invariable, the times of vibration must 
lutely equal at the same place. 




Fig. 107. 

Lispension 
be abso- 

The pendulums just do 



PERIODIC MOTION. 177 

scribed are principally used for astronomical clocks, where 
great accuracy and great uniformity in the measure of time 
is indispensable. 

Basis of a system of Weights and Measures. 

128. The pendulum is of further importance, in a prac- 
tical point of view, in furnishing the standard of comparison 
which has been made use of as a basis of the English system 
of weights and measures. The length of the seconds pendu- 
lum at any place, can always be found, and it must always 
be the same at that place. We have seen that this length 
was determined, with great accuracy, in the Tower of Lon- 
don, to be 3.2616 ft. It has been decreed by the British 
Government, that the 3-.2^T^ tn P art °f tne length of the 
simple seconds pendulum, in the Tower of London, shall be 
regarded as a standard foot. From this, by multiplication 
and division, every other unit of lineal measure may be de- 
rived. By constructing squares and cubes upon the linear 
units, we at once arrive at the units of area and of volume. 

It has further been decreed, that a cubic foot of distilled 
water, at the temperature of maximum density, shall be re- 
garded as weighing 1000 standard ounces. This fixes the 
ounce ; and by multiplication and division, all other units of 
weight may be derived. 

This system enables us to refer to the original standard, 
when, from any circumstances, doubt may exist as to the 
accuracy of standard measures. Even should every vestige 
of a standard be swept from existence, they might be per- 
fectly restored, by the process above indicated. 

The American system of weights and measures is adopted 
from that of Great Britain, and is, in all respects, the same 
as that above described. 

EXAMPLES. 

1. The length of a seconds pendulum is 39.13921 iu. If 
it be shortened 0.130464 in., how many vibrations will be 
gained in a day of 24 hours? 



178 MECHANICS. 

SOLUTION. 

The times of vibration of two pendulums at the same 
place, are to each other as the square roots of their lengths 
(Eq. 97). Hence, the number of vibrations made in any- 
given time, are inversely proportional to the square roots of 
their lengths. If, therefore, we denote the number of vi- 
brations gained in 24 hours, or 86400 seconds, by a;, we 
shall have, 



86400 : 86400 + x : : -|/39.008747 : |/39. 13921 ; 
or, 86400 : 86400 + x : : 6.2457 : 6.2561. 

Whence, x = 144, nearly. Ans. 

2. A seconds pendulum being carried to the top of a 
mountain, was observed to lose 5 vibrations per day of 
86400 seconds. Required the height of the mountain, 
reckoning the radius of the earth at 4000 miles. 

solution. 
The squares of the times of vibration, at any two points, 
are inversely proportional to the forces of gravity at 
those points (Eq. 98). But the forces of gravity at the 
same points are inversely as the squares of their distances 
from the centre of the earth. Hence, the times of vibration 
are proportional to the distances of the points from the cen- 
tre of the earth ; and, consequently, the number of vibra- 
tions in any given time, as 24 hours, for example, will be 
inversely as those distances. If, therefore, we denote the 
height of the mountain in miles by sc, we shall have, 

86400 : 86405 : : 4000 : 4000 + x. 

Whence, x = f £ J£§ = 0.2315 miles, or, 1222 feet. Ans. 

3. What is the time of vibration of a pendulum whose 
length is 60 inches, when the force of gravity is reckoned at 
32^ ft? Ans. 1.2387 sec. 



PERIODIC MOTION. 179 

4. How many vibrations will a pendulum 36 inches in 
length make in one minute, the force of gravity being the 
same as before ? Am. 62.53. 

5. A pendulum is found to make 43170 vibrations in 12 
hours. How much must it be shortened that it may beat 
seconds ? 

SOLUTION. 

We shall have, as in Example 1st, 

43170 : 43200 : : V 39 - 13921 : V 39 - 13921 + x - 
Whence, x = 0.0544 in. A?is. 

6. In a certain latitude, the length of a pendulum vi- 
brating seconds is 39 inches. What is the length of a pen- 
dulum vibrating seconds, in the same latitude, at the height 
of 21000 feet above the first station, the radius of the earth 
being 3960 miles? Am. 38.9218 in. 

7. If a pendulum make 40000 vibrations in 6 hours, at 
the level of the sea, how many vibrations will it make in the 
same time, at an elevation of 10560 feet above the same 
point, the radius of the earth being 3960 miles? 

Am. 39979.8. 
Centre of Percussion. 

129. The point O, Fig. 108, is a point at which, if the 
entire mass were concentrated, and the im r 
pressed forces applied to it, the effect produced 
would be in nowise different from what actu- 
ally obtains. Were an impulse applied at this 
point, capable of generat ng a quantity of mo- / 

tion equal and directly opposed to the resul- /... 

tant of all the quantities of motion of the ' ~" 
particles of the body, at any instant, the body Fig. ios. 
would evidently be brought to a state of rest 
without imparting any shock to the axis of suspension, 
The direction of the impulse remaining the same as before, 



180 MECHANICS. 

no matter what may be its intensity, there will still be no 
shock on the axis. This point is, therefore, called the centre 
of percussion. We may then define the centre of percus- 
sion to be that point of a body restrained by an axis, at 
which, if the body be struck in a direction perpendicular to 
a plane passed through this point and the axis of suspension, 
o shock will be imparted to the axis. It is a matter of 
common observation that, if a rod held in the hand 
be struck at a certain point, the hand will not feel the 
blow, but if it be struck at any other point of its length, 
there will be a shock felt, the intensity of which will depend 
upon the intensity of the blow, and upon the distance of its 
point of application from the first point. 

Moment of Inertia. 

130. The moment of inebtia of a body with respect to 
an axis, is the algebraic sum of the products obtained by 
multiplying the mass of each elementary particle by the 
square of its distance from the axis. Denoting the moment 
of inertia with respect to any axis, by K, the mass of any 
element of the body, by m, and its distance from the axis, 
by r, we have, from the definition, 

K— 2(mr 2 ) .... (104.) 

The moment of inertia evidently varies, in the same body, 
according to the position of the axis. To investigate the 
law of variation, let AB represent any sec- 
tion of the body by a plane perpendicular 
Xp the axis ; C, the point in which this plane 
cuts the axis ; and G, the point in which it 
cuts a parallel axis through the centre of 
gravity. Let P be any element of the 
body, whose mass is m, and denote PC by r, PG by 9, and 
CG by k. 

From the triangle CPG, according to a principle of 
Trigonometry, we have, 

r 2 =: s* + # - 2sk cos CGP. 



JP/ i \s 

ell 



MOMENT OF INERTIA. 181 

Substituting in (104), and separating the terms, we have, 

K = l(ms' 2 ) -f Zfmk*) — 2Z(mskcosCGI>). 

Or, since k is constant, and 2(m) = M, the mass of the 
entire body, Ave have, 

K = 2(ms 2 ) 4- MW — 2k2(msco&CGI > ). 

But scosCGI* — GH, the lever arm of the mass r#, 
with respect to the axis through the centre of gravity. 
Hence, 2(ms cos (7 6rP), is the algebraic sum of the mo- 
ments of all the particles of the body with respect to the 
axis through the centre of gravity ; but from the principle 
of moments, this is equal to 0. Hence, 

K= 2{ms 2 ) + Mk 2 . . . (105.) 

The first term of the second member of (105), is the ex- 
pression for the moment of inertia, with respect to the axis 
through the centre of gravity. 

Hence, the moment of inertia of a body with respect to 
any axis, is equal to the moment of inertia with respect to 
a parallel axis through the centre of gravity, plus the mass 
of the body into the square of the distance between the two 
axes. 

The moment of inertia is, therefore, the least possible, 
when the axis passes through the centre of gravity. If any 
number of parallel axes be taken at equal distances from the 
centre of gravity, the moment of inertia with respect to 
each, will be the same. 

The moment of inertia of a body with respect to any 
axis, may be determined experimentally as follows. Make 
the axis horizontal, and allow the body to vibrate about it, 
as a compound pendulum. Find the time of a single vibra- 
tion, and denote it by t. This value of £, in Equation (95), 
makes known the value of I Determine the centre of 
gravity by some one of the methods given, and denote its 



182 MECHANICS. 

distance from the axis, by k. Find the mass of the bodj 
(Art. 11), and denote it by M. 
We have, from Equation (102), 

Mid = 2(?77r') = K. 

Substituting for M, I, and 7c, the values already found, 
and the value of K will be the moment of inertia, with res- 
pect to the assumed axis. Subtract from this the value of 
Mk 2 , and the remainder will be the moment of inertia 
with respect to a parallel axis through the centre of gravity. 
The moment of inertia of a homogeneous body of regular 
figure, is most readily found by means of the calculus A 
few examples of the application of the calculus to finding 
the moment of inertia of bodies are subjoined. 

Application of the Calculus to determine the Moment of Inertia. 

131. To render Formula (104) suitable to the application 
of the calculus, we have simply to change the sign of sum- 
mation, 2, to that of integration, /, and to replace m by 
dM, and r by x. This gives, 

K= ftfdM .... (106.) 

Example 1. To find the moment of inertia of a rod or bar 
of uniform thickness with respect to an axis through its 
centre of gravity and perpendicular to the length of the 
rod. 

Let AB represent the rod, G its centre of gravity, and 
E any element contained by 
planes at right angles to the 
length of the rod and infinitely 
near each other. Denote the 
mass of the rod by M, its length, 
by 21, the distance GE, by x, and Fi 110 

the thickness of the element E, 
by dx. Then will the mass of the element E be equal to 







D 


p 


J^ 




i , i | 


p 


E 




G E 






C 





MOMENT OF INERTIA. 



183 



M 

— dx. Substituting this for dM, in Equation (106), and 

At 

integrating between the limits — I and + I, we have, 

+ i 

— i 



For any parallel axis whose distance from G is d, we shall 
have, 



K' = M(^ + d^ ( 107.) 



These two formulas are entirely independent of the 
breadth of the filament in the direction of the axis DC. 
They will, therefore, hold good when the filament AJB is 
replaced by the rectangle KF. In this case, M becomes 
the mass of the rectangle, 21 the length of the rectangle, 
and d the distance of the centre of gravity of the rectangle 
from the axis parallel to one of its ends. 

Example 2. To find the moment of inertia of a thin 
circular plate about one of its diameters. 

Let A OB represent the plate, AB the axis, and CD' 
any element parallel to AJB. Denote 
the radius C, by r, the distance OE, 
by as, the breadth of the element EE, 
by dx, and its length D O, by 2y. If 
we denote the entire mass of the 
plate, by M, the mass of the element 

OB will be equal to M y 



a ' 



or, 



smce 



y 



•yV - x\ 



we have, 




dM = M-± - dx. 



irr 



Substituting in Equation (106), we have, 



184 



MECHANICS. 



K = 



M 



J vcr* v ' 



dx. 



Integrating by the aid of Formulas A and B (Integral 
Calculus), and taking the integral between the limits 
x — — r, and x = + r, we find, 



and for a parallel axis at a distance from A B equal to d, 



K' = M(j + d a ) 



(108.) 



Example 3. To find the moment of inertia of a circular 
plate with respect to an axis through 
its centre perpendicular to the face of 
the plate. 

Let the dimensions and mass of the 
plate be the same as before. Let KL 
be an elemetary ring whose radius is x, 
and whose breadth dx. Then will the 
mass of the elementary ring be equal 



to M — , or dM 



2Mxdx 



irr 



2 ? 




Substituting this in Equation (106), and taking the 
integral between the limits x = 0, and x = r, we have, 



* = /; 

o 



2Mx*dx _ Mr 2 
r 2 = ~2~~ 



For a parallel axis at a distance d from the primitive 
axis, 



K { 



>^ M (^\-d*) (109.) 



MOMENT OF INERTIA. 



185 



Example 4. To find the moment of inertia of a circular 
ring, such as maybe generated by revolving a rectangle about 
a line parallel to one of its sides, 
taken with respect to an axis through 
the centre of gravity and perpendi- 
cular to the face of the ring. This case 
differs but little from the preceding. 
Denote the inner radius by r, the 
outer radius by r\ and the mass of 
the ring by M. If we take, as before, 
an elementary ring whose radius is 
aj, and whose breadth is dx, we shall have for its mass, 




Fig. 113. 



dM = M 



2xdx 



Substituting in Equation (106), and integrating between 
the limits r, and r\ we have, 



r' 



* = / 



M 



^dx _ r'*-r* w r '* + r * 

-r*~ 2(r' 2 - r a ) 2 



For a parallel axis at a distance from the primitive axis 
equal to d, we have. 



"' = mC 



r'> + r u 



+ 



*) 



(110.) 



If in these values of K and K' we make r = 0, we shall 
deduce the results of the last example. 



Example 5. To find the moment of inertia of a right 
cylinder with respect to an axis through the centre of 
gravity and perpendicular to the axis of the cylinder. 

Let AB represent the cylinder, CD the axis through 



c 

E 



186* MECHANICS. 

its centre of gravity, and E an ele- 
ment of the cylinder between two 
planes perpendicular to the axis, and 
distant from each other, by dx. De- 
note the length of the cylinder by 2£, 
the area of its cross section by -ttt 2 , FioP 114 

r being the radius of the base; the 
distance of the section E from the centre of gravity, by x, 
and the mass of the cylinder, by M. 

dx 
The mass of the element E is equal to M— • Its 

moment of inertia with respect to its diameter parallel to 

JSIdx r 2 
CD, is equal to x — (Example 2), and with respect 

to CD parallel to it, —■ (£.+ «*»). 

Integrating this expression between the limits x = — I, 
and x = + I, we have, 



+ i 



i 



WS(r*-0* = *<r + T) 



For an axis parallel to the primitive one, and at a 
distance from it equal to d, 

JT' = Jf(j4-|+"cp) (in.) 



Centre of Gyration. 

132. The centre of gyration of a body with respect to 
an axis, is a point at which, if the entire mass be concen- 
trated, its moment of inertia will remain unchanged. The 
distance from this point to the axis is called the radius of 
gyration. 



MOMENT OF INERTIA. 187 

Let M denote the mass of the body, and k' its radius of 
gyration ; then will the moment of inertia of the concen 
trated mass with respect to the axis, be equal to M k' 2 ; but 
this must, by definition, be equal to the moment of inertia 
with respect to the same axis, or 2(mr*) ; hence, 



Mk" = 2(m- 2 ; ; or, h' = ^/-^sP . (112.) 

That is, the radius of gyration is equal to the square 
root of the quotient obtained by dividing the moment of 
inertia with respect to the same axis, by the entire mass. 

Since M is constant for the same body, it follows that the 
radius of gyration will be the least possible when the 
moment of inertia is the least possible, that is, when the 
axis passes through the centre of gravity. This minimum 
radius is called the principal radius of gyration. If we 
denote the principal radius of gyration by k, we shall have, 
from the examples of Article (131), the following results: 

fw — 

Example 1, . h' = \/— + d? ; k = I y^J. 



u» r 

Example 2, . h' = \ — ■ + d 2 ; k = - 

V 4 2 



Example 3, . h' = y "J + d * 5 * ' = rVf. 

Example^ . h' = ^" + ** + & ', *. = J- + ^ 
^^ 5, . * '= \/j + !; + &'> & = \/j + | • 



188 



MECHANICS. 



CHAPTER VI. 



CURVILINEAR AND ROTARY MOTION. 



Motion of Projectiles. 

133, If a body is projected obliquely upwards in 
vacuum, and then abandoned to the force of gravity, it will 
be continually deflected from a rectilinear path, and, after 
describing a curvilinear trajectory, will finally reach the 
horizontal plane from which it started. 

The starting point is called the point of projection / the 
distance from the point of projection to the point at which 
the projectile again reaches the horizontal plane, through 
the point of projection, is called the range, and the time 
occupied is called the time of flight. The only forces to be 
considered, are the initial im- 
pulse and the force of gravity. 
Hence, the trajectory will lie in 
a vertical plane passing through 
the line of direction of the 
initial impulse. Let GAB rep- 
resent this plane, A the point 
of projection, AH the range, 
and A a vertical line through 

A. Take AB and AG as co-ordinate axes; denote the 
angle of projection BAB, by a, and the velocity due to the 
initial impulse, by v. Resolve the velocity v into two com- 
ponents, one in the direction AG, and the other in the 
direction AB. We shall have, for the former, v sina, and, 
for the latter, v cos*. 

The velocities, and, consequently, the spaces described in 
the direction of the co-ordinate axes, will (Art. 18) be en- 
tirely independent of each other. Penote the space 




Pig. 115, 



OURVILINEAR AND ROTARY MOTION. 189 

described in the direction A 0, in any arbitrary time t, by y 
The circumstances of motion in this direction, are those of a 
body projected vertically upwards with an initial velocity 
v sina, and then continually acted upon by the force of 
gravity. Hence, Equation (18) is applicable. Making, in 
that equation, h — y, and v' = v sina, we have, 

y = -ysinatf — \gf . . . . (113.) 

Denote the space described in the direction of the axis 
AB, in any arbitrary time t, by x. The only force acting 
in the direction of this axis, is the component of the initial 
impulse. Hence, the motion in the direction of the axis of 
x will be uniform, and Equation (55) is applicable. Making 
s = x, and v = v cosa, we have, 

x = v cosa t (114.) 

If we suppose t to be the same in Equations (113) and 
(114), they will be simultaneous, and, taken together, will 
make known the position of the projectile at any instant. 

From (114), we have, 



t = 



V COSa 



which, substituted in (113), gives, 



sina ax* , 

y = — g- B f . « • • (H5.) 
cos* 2?rcos a y 



an equation which is entirely independent of t. It, there- 
fore, expresses the relation between x and y for any value 
of t whatever, and is, consequently, the equation of the tra- 
jectory. Equation (115) is the equation of a parabola 
whose axis is vertical. Hence, the required trajectory is a 
parabola. 



190 MECHANICS. 

To find an expression for the range, make y = 0, in (115), 

and deduce the corresponding value of x. Placing the value 

of y equal to 0, we have, 

sina ax* 

jg ** , _ o • 

cosa 2?; 2 cos 2 a 

2v 2 sina cosa 

.*. x = 0, and x = • 

9 

The first value of a? corresponds to the point of projection, 
and the second is the value of the range, AB. 
From trigonometry, we have, 

2sina cosa = sin2a. 

If we denote the height due to the initial velocity, by A, 
we shall have, 

v 2 = 2gh. 

Substituting these in the second value of jc, and denotmg 
the range by r, we have, 

r = 2Asin2a ( 116.) 

The greatest value of r will correspond to the value 
a = 45°, in which case, 2a = 90°, and sin 2a = 1. 
Hence, we have, for the greatest range, 

r = 2h. 

That is, it is equal to twice, the height due to the initio* 
velocity. 

If, in (116), we replace a by 90° — a, we shall have, 

r — 2Asin(]80° —2a) = 2h sin2a, 

the same value as before. Hence, we conclude that there 
are two angles of projection, complements of each other, 



CURVILINEAR AND ROTARY MOTION. 



191 



which give the same range. The trajectories in the two 
cases are not the same, as may be shown by substituting the 
values of a, and 90° — a, in Equation (115). The greater 
angle of projection gives a higher elevation, and, conse- 
quently, the projectile descends more vertically. It is for 
this reason that the gunner selects the greater of the two 
angles of elevation when he desires to crush an object, 
and the lesser one when he desires to batter, or overturn 
the object. If a = 90°, the value of r becomes 0. That 
is, if a body be projected vertically upwards, it will return 
to the point of projection. 

To find the time of flight, make x = r, in Equation (114), 
and deduce the corresponding value of t. This gives, 



t = 



VCOSa 



(117. 



The range being the same, the time of flight will be 
greatest when a is greatest. Equation (114) also gives the 
time required for the body to describe any distance in the 
direction of the horizontal line AB. 

In Equation (117) there are four quantities, t, r, v, and a, 
and from it, if any three are given, the remaining one may 
be determined. 

As an application of the principles just deduced, let it be 
required to determine the angle 
of projection, in order that the 
projectile may strike a point 
5j at a horizontal distance 
AG — x' from the point of 
projection, and at a height 
(r/I = y' above it. -A- 

Since the point H lies on the Fig. u«. 

trajectory, its co-ordinates must 
satisfy the equation of the curve, giving 




y' = a' tana 



g* 



2w a C08*a 



192 MECHANICS. 

From trigonometry, we have, 

1 1 



cos'a = 



sec 2 a 1 -f- tan 2 a 



Substituting this in the preceding equation, we havew 
after clearing of fractions, 

Wy' = 2vVtana — gx t2 (l -f tan 2 a) ; 

or, transposing and reducing, 



tanV - ^ tarn = ■ W + ^ 



gx gx' 2 



Hence, 



yx' ± V #V 3 
or, making v* = 2gh, 



2v' 2 g f + gx' 2 



tana = 

gx v g-x" gx 



2h r /4A 2 4%'+a' 2 2A =fc t/4A 8 — 4hy —x' 

tana = — r ± \ / — — — * 2 

#' V a?' 2 jc' 2 »' 



This shows that there are, in general, two angles of pro- 
jection, under either of which the point may be struct. 
If we suppose 

x n = 4A 2 - 4%' .... (118.) 

the quantity under the radical sign will be 0, and the two 
angles of projection will become one. 

But if x' and y' be regarded as variables, Equation (118) 
represents a parabola whose axis is a vertical passing 
through the point of projection. Its vertex is at a distance 
above the point A, equal to A, its focus is at A, and its 
parameter is equal to 4A, or twice the range. 

If we suppose 

x' 2 < 4A 2 — 4%', 



CURVILINEAR AND ROTARY MOTION. 



193 



the point (x\ y'), will lie within the parabola just described, 
the quantity under the radical sign will be positive, and 
there will be two real values of tan a, and, consequently, 
two angles of projection, under either of which the point 
may be struck. 
If we suppose 

x n > 4A 2 - 4Ay', 



the point (x\ y'), will be without this parabola, the values 
of tana will both be imaginary, and there will be no angle 
under which the point can be struck. 




B' 



211 



Fig. 117. 

Let the parabola B'LB represent the curve whose equa- 
tion is 

x n = 4A 2 — 4/iy'. 

Conceive it to be revolved about A JO, as an axis generat- 
ing a paraboloid of revolution. Then, from what has preced- 
ed, we conclude, first, that every point lying within the 
surface may be reached from A, with a given initial velocity, 
under two different angles of projection ; second, that every 
point lying on the surface can be reached, but only by a sin- 
gle angle of projection ; thirdly, that no point lying without 
the surface can be reached at all. 

If we suppose a body to be projected horizontally from an 
elevated point A, the trajectory will be 
made known by Equation (115) by sim- 
ply making a = ; whence, sina = 0, 
and cosx = 1. Substituting and reduc- 
ing, we have, 



V = 



9 X 
2v* 



(119.) 




194 MECHANICS. 

For every value of x, y is negative, which shows that 
every point of the trajectory lies below the horizontal line 
through the point of projection. If we suppose ordinates to 
be estimated positively downwards, we shall have, 

y=g • • • • (120.) 

To find the point at which the trajectory will reach any 
horizontal plane B (7, whose distance below the point A is 
A', we make y = A', in (120), whence, 

x = JBG = v\/ . . . (121.) 

On account of the resistance of the air, the results of the 
preceding 'discussion will be greatly modified. They will, 
however, approach more nearly to the observed phenomena, 
as the velocity is diminished and the density of the projec- 
tile increased. The atmospheric resistance increases as the 
square of the velocity, and as the cross section of the pro- 
jectile exposed to the action of the resistance. In the air, 
it is found that, under ordinary circumstances, the maximum 
range is obtained by an angle of projection not far from 
34°. 

EXAMPLE S. 

1. What is the time of flight of a projectile, when the 
angle of projection is 45°, and the range 6000 feet? 

SOLUTION. 

When the angle of projection is 45°, the range is equal to 
twice the height due to the velocity of projection. Denot- 
ing this velocity by v, we shall have, 

w 3 = 2gh — 2 X 32 i X 0000 — 193000. 



CURVILINEAR AND ROTARY MOTION. 195 

Whence, we find, 

v = 439.3 ft. 

From Equation (117), we have, 

= 19.3 sec. Ans. 



vcosol 439.3 cos45 c 

2. What is the range of a projectile, when the angle of 
projection is 30°, and the initial velocity 200 feet ? 

Ans. 1076.9 ft. 

3. The angle of projection under which a shell is thrown 
is 32°, and the range 3250 feet. What is the time of flight ? 

Ans. 11.25 sec, nearly. 

4. Find the angle of projection and velocity of projec- 
tion of a shell, so that its trajectory shall pass through two 
points, the co-ordinates of the first being x = 1700 ft., 
y = 10 ft., and of the second, x = 1800 ft., y = 10 ft. 

SOLUTION. 

Substituting for x and y, in Equation (115), (1700, 10), 
and (1800, 10), we have, 

10 = 1700tana-i»; 

2v cos cc 

and, 

(1800)V 



10 = 1800tana — 



2v 2 cos 2 a 



Finding the value of a a from each of these equa- 

Z/V COS CC 

tions, and placing the two equal to each other, we have, 
after reduction, 

(18) 2 (l-170tana) - (I7) 2 (l-180tana). 



196 MECHANICS. 

Whence, by solution, 

tana = g^- = 0.01144, nearly ; /. a = 39' 19". 
We have, from trigonometry, 

1 1 374544 - .. 

cos'a = = — = — = .99987. 

sec a a l+tan 2 a 374593 

Substituting for tana and cosa in the first equation 
their values as just deduced, we find, for i> 2 , 

„. = (■^°°)V = 92961666^ 

2cos 2 a(1700tana— 10) 18.89 

Whence, 

V = 22-18.3 ft. 

The required angle of projection is, therefore, 39'19", and 
the required initial velocity, 2218.3 ft. 

4. At what elevation must a shell be projected with a 
relocity of 400 feet, that it may range 7500 feet on a plane 
which descends at an angle of 30 ? 

SOLUTION. 

The co-ordinates of the point at which the shell strikes, are 

x' = 7500cos30° = 6495 ; and y' = — 7500sin30° = — 3750. 

And denoting the height due to the velocity 400 ft., by h t 
we have, 

h = ~ = 2486 ft. 
Substituting these values in the formula, 



2A ± -v/4A 2 — 4Ay ; — x' % 
tana — ; : , 



CURVILINEAR AND ROTARY MOTION. 197 

and reducing, we have, 

4972 ± 4453 

tana = • 

6495 

Hence, a = 4° 34' 10", and 55° 25' 41". Am. 

Centripetal and Centrifugal Forces. 

134. Curvilinear motion can only result from the action 
of an incessant force, whose direction differs from that of 
the original impulse. This force is called the deflecting 
force, and may arise from one or more active forces, or it 
may result from the resistance offered by a rigid body, as 
when a ball is compelled to run in a curved greove. What- 
ever may be the nature of the deflecting forces, we can 
always conceive them to be replaced by a single incessant 
force acting transversely to the path of the body. Let the 
deflecting force be resolved into two components, one nor- 
mal to the path of the body, and the other tangential to it. 
The latter force will act to accelerate or retard the motion 
of the body, according to the direction of the deflecting 
force ; the former alone is effective in changing the direction 
of the motion. The normal component is always directed 
towards the concave side of the curve, and is called the 
centripetal force. The body resists this force, by virtue of 
its inertia, and, from the law of inertia, the resistance must 
be equal and directly opposed to the centripetal force. This 
force of resistance is called the centrifugal force. Hence, 
we may define the centrifugal force to be the resistance 
xohich a body offers to a force which tends to deflect it from 
a rectilineal path. The centripetal and centrifugal forces 
taken together, are called central forces. 

Measure of the Centrifugal Force. 

135. To deduce an expression for the measure of the 
centrifugal force, let us first consider the case of a single 
material point, which is constrained to move in a circular 



J98 



MECHANICS. 



path by a force constantly directed towards the centre, as 
when a solid body is confined by a string and whirled around 
a fixed point. In this case, the tangential component of the 
deflecting force is always 0. There will be no loss of velo- 
city in consequence of a change of direction in the motion 
(Art. 120). Hence, the motion of the point will be uniform. 
Let ABB represent the path of the body, and V its 
centre. Suppose the circumference 
of the circle to be a regular polygon, 
having an infinite number of sides, of 
which AB is one; and denote each 
of these sides by ds. When the body 
reaches A, it tends, by virtue of its 
inertia, to move in the direction of the 
tangent AT; but, in consequence of 
the action of the centripetal force di- 
rected towards V, it is constrained to 
describe the side ds in the time dt. If 

we draw B C parallel to A T, it will be perpendicular to the 
diameter AD, and AC will represent the space through 
which the body has been drawn from the tangent, in the 
time dt. If we denote the acceleration due to the centripetal 
force by/, and suppose it to be constant during the time dt, 
we shall have, from Art. 114, 




AC 



\fdf 



(122. 



From a property of right-angled triangles, we have, since 
AB = ds, 

ds' = AC X AB ; or, ds' = AC X 2r. 
Whence, 



AG = 



~2r 



Substituting this value of AC in (122), and solving with 
respect to /, 

1 - dP r 



CURVILINEAR AND ROTARY MOTION. 199 

ds 2 

But — = v 2 (Art. 113), in which v denotes the velocity 
(it 
of the moving point. Substituting in the preceding equa- 
tion, we have, 

f=~ (123.) 



Here f is the acceleration due to the deflecting force ; 
and, since this is exactly equal to the centrifugal force, we 
have the acceleration due to the centrifugal force equal to 
the square of the velocity, divided by the radius of the 
circle. 

If the mass of the body be denoted by 3f, and the entire 
centrifugal force by F, we shall have (Art. 24), 

F=^ (124.) 

r 

If we suppose the body to be moving on any curve what- 
ever, we may, whilst it is passing over any two consecutive 
elements, regard it as moving on the arc of the osculatory 
circle to the curve which contains these elements ; and, fur- 
ther, we may regard the velocity as uniform during the 
infinitely small time required to describe these elements. 
The direction of the centrifugal force being normal to the 
curve, must pass through the centre of the osculatory circle. 
Hence, all the circumstances of motion are the same as 
before, and Equations (123) and (124) will be applicable, 
provided r be taken as the radius of the curvature. Hence, 
we may enunciate the law of the centrifugal force as 
follows : 

The acceleration due to the centrifugal force is equal to 
the square of the velocity of the body divided by the radius 
of curvature. 

The entire centrifugal force is equal to the acceleration, 
multiplied by the mass of the body. 

In the case of a body whirled around a centre, and re- 
strained by a string, the tension of the string, or the force 



200 



MECHANICS. 



exerted to bicak it, will be measured by the centrifugal 
force. The radius remaining constant, the tension will 
increase as the square of the velocity. 

Centrifugal Force at points of the Earth's Surface. 

136. Let it be required to determine the centrifugal 
force at different points of the earth's surface, due to its 
rotation on its axis. 

Suppose the earth spherical. Let A be any point on the 
surface, PQP a meridian 
section through A, PP the 
axis, FQ the equator, and 
AB perpendicular to PP\ 
the radius of the parallel of 
latitude through A. Denote 
the radius of the earth by r, 
the radius of the parallel 
through A by r\ and the 
latitude of A, or the angle 
ACQ, by I The time of 
revolution being the same for every point on the earth's 
surface, the velocities of Q and A will be to each other as 
their distances from the axis. Denoting these velocities by 
v and v\ we have, 




v : v 



whence, 



vr 



But, from the right-angled triangle CAP, since the angle 
at A is equal to 2, we have, 

r' = rcosl. 



Substituting this value of r' in the value of v\ and re 
ducing, we have, 

v' — v cosZ. 



CURVILINEAR AND ROTARY MOTION. 201 

If we denote the acceleration due to the centrifugal force 
at the equator by/*, we shall have, Equation (123), 

/ =- (125-) 

In like manner, if we denote the acceleration due to the 
centrifugal force at A, by/', we shall have, 

v'* 

Substituting for v' and r' their values, previously deduced, 
we get, 

/<=^ (126.) 

Comparing Equations (125) and (126), we find, 

/:/':: 1 : cosZ, .% /' =/cos* . (127.) 

That is, the centrifugal force at any point on the earth's 
surface is equal to the centrifugal force at the equator, 
multiplied by the cosine of the latitude of the place. 

Let AE, perpendicular to PP\ represent the value of 
/', and resolve it into two components, one tangential, and 
the other normal to the meridian section. Prolong CA, and 
draw AD perpendicular to it at A. Complete the rectangle 
FD on AE as a diagonal. Then will AD represent the 
tangential, and AF the normal component of f. In the 
right-angled triangle AFE, the angle at A is equal to I. 
Hence, 

FE = AD = /'sin/ = fcoslsinl = ^— . ( 128.) 

AF = f'cosl = /cosV . . . . ( 129.) 

From (128), we conclude that the tangential component is 
9* 



202 MECHANIC*. 

at the equator, goes on increasing till I = 45°, where it 
is a maximum ; then goes on decreasing till the latitude is 
90° when it again becomes 0. 

The effect of the tangential component is to heap up the 
particles of the earth about the equator, and, were the 
earth in a fluid state, this process would go on till the effect 
of the tangential component was exactly counterbalanced 
by component of gravity acting down the inclined plane 
thus found, when the particles would be in a state of equili- 
brium. The higher analysis has shown that the form of 
equilibrium is that of an oblate spheroid, differing but 
slightly from that which our globe is found to possess by 
actual measurement. 

From Equation (129), we see that the normal component 
of the centrifugal force is equal to the centrifugal force at 
the equator multiplied by the square of the cosine of the 
latitude of the place. 

This component is directly opposed to gravity, and, con- 
sequently, tends to diminish the weight of all bodies on the 
surface of the earth. The value of this component is 
greatest at the equator, and diminishes towards the poles, 
where it becomes equal to 0. From the action of the 
normal component of the centrifugal force, and from the 
flattened form of the earth due to the tangential component 
bringing the polar regions nearer the centre of the earth, 
the measured force of gravity ought to increase in passing 
from the equator towards the poles. This is found, by 
observation, to be the case. 

The radius of the earth at the equator is found, by 
measurement, to be about 3962.8 miles, which, multiplied by 
2 i 7i', will give the entire circumference of the equator. If 
this be divided by the number of seconds in a day, 86400, 
we find the value of v. Substituting this value of v and 
that of r just given, in Equation (125), we should find, 

/ = 0.1112 ft., 

for the measure of the centrifugal force at the equator. If 



CURVILINEAR AND KOTART MOTION. 203 

this be multiplied by the square of the cosine of the latitude, 
of any place, we shall have the value of the normal com- 
ponent of the centrifugal force at that place. 

Centrifugal Force of Extended Masses. 

136. We have supposed, in what precedes, the dimen 
sions of the body under consideration to be extremely small ; 
let us next examine the case of a body, of any dimensions 
whatever, constrained to revolve about a fixed axis, with 
which it is invariably connected. If we suppose this body 
to be divided into infinitely small elements, whose directions 
are parallel to the axis, the centrifugal force of each element 
will, from what has preceded, be equal to the mass of the 
element into the square of its velocity, divided by its dis- 
tance from the axis. If a plane be passed through the cen- 
tre of gravity of the body, perpendicular to the axis, we 
may, without impairing the generality of the result, suppose 
the mass of each element to be concentrated at the point in 
which this plane cuts the line of direction of the element. 

Let XGY be the plane through the centre of gravity of 
the body perpendicular to the axis of 

revolution, AB the section cut out a/" X 

of the body, or the projection of the 
body on the plane, and C the point 
in which it cuts the axis. Take C as 
the origin of a system of rectangular 
axes, and let CX be the axis of JT, 
CY the axis of Y, and let m be the Fig. 121 

point at which the mass of one of these 

filaments is concentrated, and denote that mass by m. De- 
note the co-ordinates of m by x and y, its distance from 
C by r, and its velocity by v. The centrifugaj force of tha 
mass «i will be equal to 



If we denote the angular velocity of the body by F 7 , rhe 



^04 MECHANICS. 

velocity of the point m will be equal to r J 77 , which, being 
substituted in the expression for the centrifugal force just 
deduced, gives 

mrV 72 . 

Let this force be resolved into two components, respec- 
tively parallel to the axes GX and G Y, We shall have, 
for these components, the expressions, 

mr V r2 cosm CX, and mr "F^sinm GX. 

But from the figure, we have, 

X 1/ 

cosm GX = - , and sinm GX = - • 
r r 

Substituting these values in the preceding expressions, 
and reducing, we have, for the two components, 

mx I 7 "' 2 , and my V 2 . 

In like manner, if we denote the masses of the remaining 
filaments by m r , m'\ &c, the co-ordinates of the points at 
which they are cut by the plane XCY, by x\ y' ; x", y", 
&c, their distances from the axis by r', r" , &c, and resolve 
the centrifugal forces into components, respectively parallel 
to the axes, we shall have, since V remains the same, 

m' x' V'\ m' y' V' 2 ; 

m"x"V'\ m"y"V n \ 

<fcc, &c. 

If we denote the sum of the components in the direction 
of the axis of X by X, and in the direction of the axis 
of Y by Y, we shall have, 

X= -Z(mx)V'\ and F=2(»jy)F a . 



CURVILINEAR AND ROTARY MOTION. 205 

If, now, we denote the entire mass of the body, by M, 
and suppose it concentrated at its centre of gravity 0, 
whose co-ordinates are designated by x x , and y x , and whose 
distance from C is equal to r v we shall have, from the 
principle of the centre of gravity (Art. 51), 

2(mx) = Mx^ and 2 (my) = My x . 

Substituting above, we have, 

X = M V'*x» and Y = M F'^. 

If we denote the resultant of all the centrifugal forces, 
which will be the centrifugal force of the body, by R, we 
shall have, 



R =^X* + Y> = MV*^x* + Vi = MV"r v 

But if the velocity of the centre of gravity be denoted by 
T 7 *, we shall have, 

F=Tr l5 or, V"=^; 

which, substituted in the preceding result, gives, for the 
resultant, 

R^ 1 ^— (130.) 

i\ 

The line of direction of R is made known by the equa- 
tions, 

X Y 

cosa = — , and cos& = — ; 
r, r x 

it, therefore, passes through the centre of gravity 0. 

Hence, we conclude, that the centrifugal force of an ex- 
tended mass, constrained to revolve about a fixed axis, loith 
which it is invariably connected, is the same as though the 
entire mass were concentrated at its centre of gravity. 



206 MECHANICS. 



Pressure on the Axis. 



137. The centrifugal force, passing through the centre 
of gravity and intersecting the axis, will exert its entire 
effect in creating a pressure upon the axis of revolution. 
By inspecting the equation, 

B =: MV n r v 

we see that this pressure will increase with the mass, the 
angular velocity, and the distance of the centre of gravity 
from the axis. When the last distance is 0, that is, when 
the axis of revolution passes through the centre of gravity, 
there will be no pressure on the axis arising from the centri- 
fugal force, no matter what may be the mass of the body or 
its angular velocity. Such is the case of the earth revolving 
on its axis. 

Principal Axes. 

138. Suppose the axis about which a body revolves to 
become free, so that the body can move in any direction. 
If that axis be not one of symmetry, it will be pressed un- 
equally in different directions by the centrifugal force, and 
will immediately alter its position. The body will for an 
mstant rotate about some other line, which will immediately 
change its position, giving place to a new axis of rotation, 
which will instantly change its position, and so on, until an 
axis is reached which is pressed equally in all directions by 
the centrifugal forces of the elements. The body will then 
continue to revolve about this line, by virtue of its inertia, 
until the revolution is destroyed by the action of some 
extraneous force. Such an axis is called a principal axis 
of rotation. Every body has at least one such axis, and 
may have more. The axis of a cone or cylinder is a prin- 
cipal axis j any diameter of a sphere is a principal axis / in 
short, any axis of symmetry of a homogeneous solid is a 
principal axis. The shortest axis of an oblate spheroid is 
a principal axis; and it is found by observation that all of 
the planets of the solar system, which are oblate spheroids, 



CURVILINEAR AND ROTARY MOTION. 207 

revolve about their shorter axes, whatever may be the incli- 
nation of these axes to the planes of their orbits. Were 
the earth, by the action of any extraneous force, constrained 
to revolve about some other axis than that about which it is 
found to revolve, it would, as soon as the force ceased to 
act, return to its present axis of rotation. 

Experimental Illustrations. 

139. The principles relating to the centrifugal force 
admit of experimental illustration. The instrument repre- 
sented in the figure, may be employed to show the value of 
the centrifugal force. A repre- 
sents a vertical axle upon which -RikcmrS . 2 
is mounted a wheel E, commu- 
nicating with a train of wheel- 
work, by means of which the __G JEr 






axle may be made to revolve 

with any angular velocity. At F . 122 

the upper end of the axle is a 

forked branch BC, sustaining a stretched wire. D and E 

are two balls which are pierced by the wire, and are free to 

move along it. Between B and E is a spiral spring, whose 

axis coincides with the wire. 

Immediately below the spring, on the horizontal part of 
the fork, is a scale for determining the distance of the ball 
E, from the axis, and for measuring the degree of compres- 
sion of the spring. Before using the instrument, the force 
required to produce any degree of compression of the 
spring is determined experimentally, and marked on the 
scale. 

If now a motion of rotation be communicated to the axis, 
the ball D will at once recede to (7, but the ball E will be 
restrained by the spiral spring. As the velocity of rotation 
is increased, the spring will be compressed more and more, 
and the ball E, will approach B. By a suitable arrange- 
ment of the wheelwork, the angular velocity of the axis 
corresponding to any degree of compression may be ascer- 



208 MECHANICS. 

tained. We have thus all the data necessary to a verifier 
tion of the law of the centrifugal force. 

If a vessel of water be made to revolve about a vertical 
axis, the interior particles will recede from the axis on 
account of the centrifugal force, and will be heaped up about 
the sides of the vessel, imparting a concave form to the 
upper surface. The concavity will become greater as the 
angular velocity is increased. 

If a circular hoop of flexible metal be fastened so that 
one of its diameters shall coincide with the axis of a 
whirling machine, its lower point being fastened to the 
horizontal beam, and a motion of rotation be imparted, the 
portions of the hoop farthest from the axis will be most 
affected by the centrifugal force, and the hoop will be 
observed to assume an elliptical form. 

If a sponge, filled with water, be attached to one of the 
arms of a whirling machine, and a motion of rotation be 
imparted, the water will be thrown from the sponge. This 
principle has been made use of in a machine for drying 
clothes. An annular trough of copper is mounted upon an 
axis by means of radial arms, the axis being connected with 
a train of wheelwork, by means of which it may be put in 
motion. The outer wall is pierced with holes for the escape 
of the water, and a lid serves to confine the articles to be 
dried. To use this instrument, the linen, after being 
washed, is placed in the annular space, and a rapid motion 
of rotation imparted to the machine. The linen is thrown, 
by the centrifugal force, against the outer wall of the instru- 
ment, and the water, being partially squeezed out, and par- 
tially thrown off by the centrifugal force, escapes through 
the holes made for the purpose. Sometimes as many as 
1,500 revolutions per minute are given to the drying 
machine, in which case, the drying process is very rapid and 
very perfect. 

If a body be whirled about an axis w T ith sufficient velo- 
city, it may happen that the centrifugal force generated 
will be greater than the force of cohesion which binds the 



CURVILINEAR AND ROTARY MOTION. 209 

particles together, in which case, the body will be torn 
asunder. It is a common occurrence that large grindstones, 
when put into a state of rapid rotation, burst, the fragments 
being thrown with great velocity away from the axis, and 
often producing much destruction. 

When a wagon, or carriage, is driven rapidly around a 
corner, or is forced to turn about a circular track, the cen- 
trifugal force generated is often sufficient to throw out the 
loose articles from the vehicle, and even to overthrow the 
vehicle itself. When a car upon a railroad track is forced 
to turn around a sharp curve, the centrifugal force generated, 
tends to throw the weight of the cars against the rail, pro- 
ducing a great amount of friction, and contributing to wear 
out both the track and the car. To obviate this difficulty 
in a measure, it is customary to raise the outer rail, so that 
the resultant of the centrifugal force, and the force of grav- 
ity, shall be sensibly perpendicular to the plane of the two 

rails. 

Elevation of the outer rail of a curved track. 

140. To find the inclination of the track, that is, the 
elevation of the outer rail, so that the resultant of the 
weight and centrifugal force 
may be perpendicular to the ^r 

line loining the two rails. Let /~^\ 

G be the centre of gravity f \ 

of the car, and let the figure 1 \ 

represent a vertical section n\\ |f:_jjl\ 

through the centre of gravity 1 wiiP^hI 

and the centre of the curved 1 B«c % — D 

track. Let GA, parallel to E . Fio , 12g -^ 

the horizon, represent the ac- 
celeration due to the centrifugal force, and GB, perpen- 
dicular to the horizon, the acceleration due to the weight 
of the car. Construct the resultant GC, of these forces, 
then must the line DE be perpendicular to GO. Denote 
the velocity of the car, by v, and the radius of the curved 
track, by r. The acceleration due to the weight will be 



210 MECHANICS. 

equal to g, the force of gravity, and the acceleration due to 

v 2 
the centrifugal force will be equal to — The tangent of the 

CB r 

angle CGB will be equal to -^tb > or > denoting the angle 

by a, we shall have, 

CB tf 
tana = -— = - — 
GB gr 

But the angle DEF is equal to the angle CGB. Denot- 
ing the distance between the rails, by d, and the elevation 
of the outer rail above the inner one, by h, we shall have, 



tana = — , very nearly. 
Equating the two values of tana, we have, 

-7=—', .". h = 1 , (131.) 

a gr gr v ' 

Hence, the elevation of the outer rail varies as the square 
of the velocity directly, and as the radius of the curve 
inversely. 

It is obvious that this correction would require to be 
different for different velocities, which, from the nature of 
the case, would be manifestly impossible. The correction 
is, therefore, made for some assumed velocity, and then 
such a form is given to the tire of the wheels as will com- 
plete the correction for different velocities. 

The Conical Pendulum. 

141. The conical pendulum consists of a solid ball at- 
tached to one end of a rod, the other end of which is con- 
nected, by means of a hinge-joint, with a vertical axle. 
When the axle is put in motion, the centrifugal force gene- 
rated in the ball causes it to recede from the axis, until an 
equilibrium is established between the weight of the ball, the 
centrifugal force, and the tension of the connecting rod. 



CURVILINEAR AND ROTARY MOTION. 211 

When the velocity is constant, the centrifugal force will be 
constant, and the centre of the ball will describe a horizontal 
circle, whose radius will depend upon the velocity. Let it 
be required to determine the time of revolution. 

Let JBD be the vertical axis, A the ball, B the hinge- 
joint, and AB the connecting rod, whose 
mass is so small, that it may be neglected, 
in comparison with that of the ball. 

Denote the required time of revolution, 
by £, the length of the arm, by £, the accele- 
ration due to the centrifugal force, by/*, and 
the angle ABC, by p. Draw AC perpen- 
dicular to BD, and denote A (7, by r, and 
BG,hyh. • — — 

From the triangle AB (7, we have, r = feinp ; and since 
r is the radius of the circle described by A, we have the 
distance passed over by A, in the time t, equal to 
2<xr = 2*rfein<p. Denoting the velocity of A, by v, we have, 
from Equation (55), 

v = 1. 



But the centrifugal force is equal to the square of the 
velocity, divided by the radius ; hence, 

j, 4<7r 2 £sin<p 
/= — ^ .... (132.) 

The forces which act upon A, are the centrifugal force in 
the direction AF, the force of gravity in the direction A G, 
and the tension of the connecting rod in the direction AB. 
In order that the ball may remain at an invariable distance 
from the axis, these three forces must be in equilibrium. 
Hence (Art. 35), 

g :f: : smBAJF 1 : : smBAG ; 

but, siniL4i^= sin(90° + <p) — cosp; 



919, 



MECHANICS. 

and, smBAG == sin(180° — (p) = sin(p ; 

whence, by substitution, 



g : f : : costp : sinp, 



9=f 



COS<p 

sincp 



Substituting for f its value, taken from (132), we have, 

4tf 2 £cosa> 
9 = — fi 

But, from the triangle AB G, we have, Zcoscp = A, wl ch 
gives, 



9 = 









( 133.) 



That is, the time of a revolution is equal to the time of a 
double vibration of a pendulum whose length is h. 

The Governor. 

142. The principle of the conical pendulum is employed 
in the governor, a machine attached to engines, to regulate 
the motive force. 

AB is a vertical axis connected with the machine near its 
working point, and revolving with a 
velocity proportional to that of the 
working point ; FE and GD are two 
arms turning freely about AB, and 
bearing heavy balls D and E, at their 
extremities ; these bars are united by 
hinge-joints with two other bars at 
G and E, these bars are also attach- 
ed to a ring at J5T, free to slide up and 
down the shaft. 

The governor is so constructed, 
that the figure GGFH is always a parallelogram. The 
ring at ^Tis connected with a lever HK, which maybe made 
to act upon the valve that admits steam to the cylinder. 




CURVILINEAR AND ROTARY MOTION'. 213 

When the shaft revolves, the centrifugal force developed 
in the balls, causes them to recede from the axis, and the 
ring H is depressed ; and when the velocity has become 
sufficiently great, the lever begins to act in closing the valve. 
If the velocity slackens, the balls approach the axis, and the 
ring H ascends, opening the valve again. In any given 
case, if we know the velocity required at the working point, 
we can from it compute the required angular velocity of the 
shaft, and, consequently, the value of t. This value of t 
being substituted in Equation (133), makes known the value 
of h. We may, therefore, make the proper adaptation of 
the ring, and of the lever HK. 

EXAMPLES. 

1. A ball weighing 10 lbs. is whirled around in a cn*cle 
whose radius is 10 feet, with a velocity of 30 feet per second. 
What is the acceleration of the centrifugal force ? 

Ans. 90 ft. 

2. In the preceding example, what is the tension upon 
the cord which restrains the ball ? 

solution. 

Denote the tension in pounds, by t ; then, since the pres- 
sures produced by two forces are proportional to their 
accelerations, we shall have, 

10 : t : : g : 90, .-. t — 28 lbs., nearly. Ans. 

3. A body is whirled around in a circular path whose 
radius is 5 feet, and it is observed that the pressure due to 
the centrifugal force is just equal to the weight of the body. 
What is the velocity of the moving body ? 

SOLUTION. 

Denoting the velocity by v, we have the acceleration 
due to the centrifugal force equal to — ; but, by the condi- 



214 MECHANICS. 

tions of the problem, this is equal to the acceleration due to 
the weight of the body. Hence, 

— = g = 32i, .-. v _ 12.7 ft. Ans. 

o 

4. In how many seconds must the earth revolve on its 
axis in order that the centrifugal force at the equator may 
exactly counterbalance the force of gravity, the radius of 
the equator being taken equal to 3962.8 miles ? 

SOLUTION. 

Reducing the miles to feet, and denoting the required 
velocity, by v 9 we have, 



20923584 



= 321, ... v — -/321 x 20923584. 



But the time of revolution is equal to the circumference 
of the equator, divided by the velocity. Denoting the time 
by t, we have, 



_ 2tf X 20923584 



and, substituting the value of w, taken from the preceding 
equation, we have, after reduction, 



2-7r</20923584 2* X 4574 - . rt 

t = — - — — — == — — ~r = 5068 sees. Ans. 

^/32i 5.67 

But the earth actually revolves in 86400 sideral, or in 
about 86164 mean solar seconds. Hence, the earth would 
have to revolve 17 times as fast as at present, in order that 
the centrifugal force at the equator might be equal to the 
force of gravity. 

5. A body is placed on a horizontal plane, which is 
made to revolve about a vertical axis, with an angular 



CUE VILINEAR AND ROTARY MOTION. 215 

velocity of 2 feet. How far must the body be situated from 
the axis that it may be on the point of sliding outwards, the 
coefficient of friction between the body and plane being 
equal to .6 ? 

SOLUTION. 

Denote the required distance by r ; then will the velocity 
of the body be equal to 2r, and the acceleration due to the 
centrifugal force will be equal to 4r. But the acceleration 
due to the force of friction is equal to 0.6 X g = 19.3 ft. 
From the conditions of the problem, these two are equal, 
hence, 

4r = 19.3 ft., .*. r = 4.825 ft. Ans. 

6. What must be the elevation of the outer rail of a rail- 
road track, the radius of curvature being 3960 ft., the 
distance between the rails 5 feet, and the velocity of the car 
30 miles per hour, in order that the centrifugal force may 
be exactly counterbalanced by the component of the weight 
parallel to the line joining the rails? 

Ans. 0.076 ft., or 0.9 in., nearly. 

7. The distance between the rails is 5 feet, the radius of 
the curve 600 feet, and the height of the centre of gravity 
of the car 5 feet. What velocity must be given to the car 
that it may be on the point of being overturned by the cen- 
trifugal force, the rails being on the same level ? 

We have, 



v — 



/o X 321 x 600 nn£ , „„- , A 

\ / s — 98 ft., or 66| m., per hour. Ans. 

V 2 x 5 4 



Work. 
143. By the term work, in mechanics, is meant the 
effect produced by a force in overcoming a resistance, such 
as weight, inertia, &c. The idea of work implies that a 
force is continually exerted, and that the point at which it 
is applied moves through a certain space. Thus, when a 
weight is raised through a vertical height, the p^wer which 



216 MECHANICS. 

overcomes the resistance offered by the weight is said to 
work, and the amount of work performed evidently depends, 
first, upon the weight raised, and, secondly, upon the 
height through which it is raised. All kinds of work may 
be assimilated to the raising of a weight. Hence it is, that- 
this kind of work is assumed as a standard to which all 
other kinds of work are referred. 

The unit of work most generally adopted in this country, 
is the effort required to raise one pound through a height 
of one foot. The number of units of work required to raise 
any weight to any height will, therefore, be equal to the. 
product obtained by multiplying the number of pounds in 
the weight by the number of feet in the height. If we 
take the weight of the body as it would be at the equator, 
for the sake of uniformity in notation, we may regard the 
weight and the mass as identical (Art. 11). If we denote 
the quantity of work expended in raising a body, by Q, the 
mass of the body, by m, and the height, by A, we shall have, 

Q = mh. 

When very large quantities of work are to be estimated, 
as in the case of steam-engines and other powerful ma- 
chines, a different unit is sometimes employed, called a 
horse power. When this unit is employed, time enters as 
an element. A horse power is a power which is capable of 
raising 33,000 lbs. through a height of one foot in one 
minute ; that is, it is a power capable of performing 33,000 
units of work in a minute of time, or 550 units of work in 
one second. When an engine, then, is spoken of as being 
of 100 horse power, it is to be understood that it is capable 
of performing 55,000 units of work in a second. 

In general, if a force acts to overcome a resistance of m 
pounds, through a distance of n feet, whatever may be the 
cause of the resistance, or whatever may be the direction 
of the motion, the quantity of work will be measured by a 
unit of work taken mn times. 



CURVILINEAR AND ROTARY MOTION. 217 

If the pressure exerted by the force is variable, we may 
conceive the path described by the point of application to 
be divided into equal parts, so small that, for each part, the 
pressure may be regarded as constant. If we denote the 
length of one of these equal parts, by p, and the force 
exerted whilst describing this path, by P, we shall have for 
the corresponding quantity of work, Pp, and for the entire 
quantity of work denoted by Q, we shall have the sum of 
these elementary quantities of work ; or, since p is the 
same for each, 

C=p2(P) (134.) 

The quotient obtained by dividing the entire quantity of 
work by the entire path, is called the mean pressure, or the 
mean resistance, and is evidently the force which, acting 
uniformly through the same path, would accomplish the 
same work. 

Work, when the power acts obliquely to the path. 

144. Let PD represent the force, and AB the path 
which the body D is constrained to 
follow. Denote the angle PDs by a, 
and suppose P\,o be resolved into two 
components, one perpendicular, and A s 3) J? 

the other parallel to AB. We shall Fis " 126 ' 

have, for the former, .Psina, and, for the latter, Pcosa. 

The former can produce no work, since, from the nature 
of the case, the point cannot move in the direction of the 
normal ; hence, the latter is the only component which 
works. Let sD be the space through which the body is 
moved in any time whatever. If we denote the pressure 
exerted in the direction of PD, by P, and the quantity of 
work, by Q, we shall have, 

Q = Pcosa x sD. 

Let fall the perpendicular ss' from s, on the direction of the 
10 



218 MECHANICS. 

force P. From the right-angled triangle Dss\ we shal] 
have, 

sD x cosa = s'P. 

Substituting this in the preceding equation, we get, 

Q = P x s'P. 

That is, the quantity of work of a force acting obliquely 
to the path along which the point of application is con- 
strained to move, is equal to the intensity of the force mul- 
tiplied by the projection of the path upon the direction of 
the force. We have supposed the intensity of the force 
P, to be expressed in pounds, or units of mass. 

If we take the distance sP, infinitely small, s'P will be 
the virtual velocity of P, and the expression for the quantity 
of work of P will be its virtual moment (Art. 38). Hence 
we say that the elementary quantity of work of a force is 
equal to its virtual moment, and, from the principle of 
virtual moments, we conclude that the algebraic sum of the 
elementary quantities of work of any number of forces 
applied at the same point, is equal to the elementary quantity 
of work of their resultant. What is true for the elementary 
quantities of work at any instant, must be equally true at 
any other instant. Hence, the algebraic sum of all the ele- 
mentary quantities of work of the components in any time 
whatever, is equal to the algebraic sum of the elementary 
quantities of work of their resultant for the same time ; that 
is, the work of the components for any time, is equal to the 
work of their resultant for the same time. This principle 
would hardly seem to require demonstration, for, from the 
very definition of a resultant, it would seem to be true of 
necessity. If the forces are in equilibrium, the entire 
quantity of work will be equal to 0. 

This principle finds an important application, in computing 
the quantity of work required to raise the material for a 
wall or building ; for raising the material from a shaft ; for 
raising water from one reservoir to another ; and a great 



CURVILINEAR AND ROTARY MOTION. 219 

variety of similar operations. In this connection, the prin- 
ciple may be enunciated as follows : The algebraic sum of 
the quantities of work required to raise the parts of a 
system through any vertical spaces, is equal to the quantity 
of work required to move the whole system over a vertical 
space equal to that described by the centre of gravity of the 
system. 

It also follows, from the same principle, that, if all the 
pieces of a machine which moves without friction be in 
equilibrium in all positions, under the action of weights 
suspended from different parts of the machine, the centre 
of gravity of the system will neither ascend nor descend 
whilst the machine is in motion. 



Work, when a body is constrained to move upon a curve. 

145. Let AB represent the curve, and suppose that the 
force is so taken that its line of direction shall 
always pass through a point P. Divide the 
curve into elements so small that each may be 
taken as a straight line, and, with Pas a 
centre, and the distances from P to the points 
of division as radii, describe arcs of circles. 
Then, denoting the force supposed constant, by 
P, we shall have (from Art. 144) the ele- 
mentary quantity of work performed whilst the 
point is moving over aa ', equal to P x ac, or P x bb'. In like 
manner, the quantity of work performed whilst the point is 
describing a' a" will be equal to P x b'b", and so on. Hence, 
by summation, we shall find the entire quantity of work 
performed in moving the body from B to A will be equal 
to P x BB' . If now we suppose the curve AB to lie in 
a vertical plane, and the force to be the force of gravity, the 
point P may be regarded as infinitely distant, the lines 
Pa, Pa' &c, will become vertical, and the lines a'b', a"b", 
will be horizontal. We may, therefore, enunciate the follow- 
ing principle : The quantity of work of the weight of a body 




220 MECHANICS. 

in descending a curve, is equal to the quantity of work of 
the same weight in descending vertically through the same 
height. This principle is immediately connected with the 
discussion in Art. 74. 

If a body in a stable position, as a pyramid resting on its 
base, be overturned by any extraneous force, the quantity 
of work will be equal to the weight of the body, multiplied 
by the vertical height to which the centre of gravity must 
be raised before reaching its highest point. This product 
might be taken as the measure of the stability of a body. 

EXAMPLES. 

1. What amount of work is required to raise 500 lbs. to 
the height of 5 yards ? Ans. 7500 units, or V500 lbs. ft. 

2. To what height can 2240 lbs. be raised by the expen- 
diture of 5600 units of work? Ans. 2.5 ft. 

3. What weight can be raised to the height of 25 feet by 
224000 units of work ? Ans. 8960 lbs. 

4. What is the effective horse power of an engine whicli 
raises 80 cubic feet of water per minute from the depth of 
360 feet, a cubic foot of water weighing 62 lbs. 

Ans. 54.11 horse power. 

5. What must be the effective horse power to raise the 
same quantity of water per minute, from a depth of 40 feet ? 

Ans. 6 horse power. 

6. How many tons of ore can be raised per hour from a 
mine 1800 feet deep, by an engine of 28 effective horse 
power, reckoning 2240 lbs. to the ton? Ans. 13£ tons. 

7. From what depth will an engine of 16 effective horse 
power raise 5 cwts. of coal per minute. 

Ans. 943 feet, nearly. 

8. In what time will an engine of 40 effective horse 
power raise 44000 cubic feet of water from a mine 360 feet 
deep, allowing 62-| pounds to the cubic foot ? 

Ans. 12 h. 30 rain. 



CURVILINEAR AND .ROTARY MOTION. 221 

9. Required the quantity of work necessary to raise the 
material for a rectangular granite wall 25 feet long, 2^ feet 
thick, and 20 feet high, the weight of granite being 162 lbs. 
per cubic foot ? 

solution. 

The weight of the wall is equal to 

162 lbs. X 25 x 2.5 X 20 = 202500 lbs. 

The height of the centre of gravity being 10 feet, the 
quantity of work is equal to 

202500 X 10 = 2025000 lbs. ft. Ans. 

10. How long would it take an engine of 4 effective 
horse power to raise the material for the wall in the last 
example? Ans. \5\ minutes, nearly. 

11. What quantity of work must be expended in drawing 
a chain from a shaft, the length of the chain being 450 feet, 
and its weight 40 lbs. to the foot ? Ans. 4050000 lbs. ft. 

12. A cylindrical well is 150 feet deep, and 10 feet in 
diameter. Supposing the well to be filled with water to the 
depth of 50 feet, how much work must be expended in 
raising it to the top, water being taken at 62.5 lbs. per 
cubic foot ? 

SOLUTION. 

The weight of the water is equal to 

« X 5 2 X 50 X 62.5 lbs. = 245437.5 lbs. 

The distance of the centre of gravity from the top is 125 
feet. Hence, the required quantity of work is equal to 

245437.5 lbs. x 125 ft. = 30679687.5 lbs. ft. Ans. 

13. What quantity of work will be required to overturn 
a right cone, with a circular base, whose altitude is 12 



222 MECHANICS. 

feet, and the radius of whose base is 4 feet, the weight of 
the material being estimated at 100 lbs. per cubic foot ? 

SOLUTION. 

The weight of the cone is equal to 

« X 4 a x 4 x 100 lbs. = 20106.24 lbs. 

If the cone turns about a tangent to its base, since the 
centre of gravity is 3 feet from the base, it will be, 



y3 2 + 4 2 = 5 feet from the tangent. 

The centre of gravity, at its highest point, will, there- 
fore, be 5 feet from the horizontal plane. It must then be 
raised 2 feet. Hence, the required quantity of work is 
equal to 

20106.24 lbs. X 2 ft. = 40212.48 lbs. ft. Ans. 

14. To show that the work required for overturning 
similar solids, similarly placed, varies as the fourth powers 
of their homologous lines. 

SOLUTION. 

Denote the altitudes of the centres of gravity, by y and ry, 
the distances from the directions of the weights to the lines 
about which they turn, by x and rx, and their weights, by 
w and r*w. 

The quantity of work required to overturn the first, 
will be, 

Q = w{Vx* + y a - 2/). 

The quantity of work required to overturn the second, 
will be, 

q' = rMv^ + ^y- r y) = r * w (V x2 + y* - y)- 

Hence, 

Q Q' : : 1 : r* . . Q.KD. 



CURVILINEAR AND ROTARY MOTION. 223 

Rotation. 

146. When a body restrained by a fixed axis, about 
which it is free to turn, is acted upon by a force, it will, in 
general, take up a motion of rotation, or revolution. In 
this kind of motion, each point of the body describes a cir- 
cle, whose centre is in the axis, and whose plane is perpen- 
dicular to the axis. The time of a complete revolution be- 
ing the same for each particle, it follows, that the velocities 
of the different particles will be proportional to their dis- 
tances from the axis. The velocity of any particle will be 
equal to its distance from the axis multiplied by the angular 
velocity (Art. 122). 

Quantity of work of a Force producing Rotation. 

147. If a force is applied obliquely to the axis of rota- 
tion, we may conceive it to be resolved into two components, 
one parallel, and the other perpendicular to the axis of rota- 
tion. The effect of the former will be counteracted by the 
resistance offered by the fixed axis ; the effect of the latter 
in producing rotation will be exactly the same as that of the 
applied force. We need, therefore, only consider those 
components whose directions are perpendicular to the axis 
of rotation. 

Let P represent any force whose line of direction is per- 
pendicular to the axis, but does 
not intersect it. Let O be the D £ 

point in which a plane through P, 
perpendicular to the axis, inter- 
sects it. Let A and C be any 
two points whatever, on the line Fig.m 

of direction of P. Suppose the 

force P to turn the system through an infinitely small angle, 
and let B and B be the new positions of A and C. Draw 
OE, Ba, and Be respectively perpendicular to PE ; draw 
also, AO, BO, CO, and BO. Denote the distances OA, 
by r, OC, by r', OE, by p, and the path described by 



,-^-A. 



224 MECHANICS. 

a point at a unit's distance from 0, by 6'. Since the angles 

A OB, and COD are equal, from 

the nature of the motion of rota- -n "B 

tion, we shall have, AB = r3 f , >, -f$r — > /^ « 

and CD — r'd' ; and since the I // y^^ 
angular motion is infinitely small, !,/>'''"' 
these lines may be regarded as 
straight lines, perpendicular re- 
spectively to OA and C. From the right-angled triangles 
ABa and CDc, we have, 

Aa == r&'cosBAa, and Go — r'&'cosDCc. 

In the right-angled triangles ABa, and OAE, we have 
.4^ perpendicular to 6M, and Aa perpendicular to OE\ 
hence, the angles BAa, and A OE, are equal, as are also 
their cosines ; hence, we have, 

cosBAa = cos^l OE == ^-« 
r 

In like manner, it may be shown, that 

cosDCc = co$COE = £. 
r 

Substituting in the equations just deduced, we have, 
Aa = p6 9 and Cc — pA ; .-. Aa — Cc ; 

whence, 

B .Aa = P. Cc = PpV. 

The first member of the equation is this quantity of work 
of JP, when its point of application is at A ; the second is 
the quantity of work of P, when its point of application is 
at C. Hence, we conclude, that the elementary quantity of 
work of a force applied to produce rotation, is always the 



CURVILINEAR AND ROTARY MOTION. 225 

same, wherever its point of application may he taken, pro- 
vided its line of direction remains unchanged. 

We conclude, also, that the elementary quantity of work 
is equal to the intensity of the force multiplied by its lever 
arm into the elementary space described by a point at a 
unit's distance from the axis. 

If we suppose the force to act for a unit of time, the 
mtensity and lever arm remaining the same, and denote the 
angular velocity, by <), we shall have, 

Q' = JPpb- 

For any number of forces similarly applied, we shall have, 

Q = 2(Fp)6 .... (135.) 

If the forces are in equilibrium, we shall have (Art. 49), 
2(Pp) = ; consequently, Q = 0. 

Hence, if any number of forces tending to produce rota- 
tion about a fixed axis, are in equilibrium, the entire quan- 
tity of work of the system of forces will be equal to 0. 

Accumulation of Work. 

148. When a body is put in motion by the action of a 
force, its inertia has to be overcome, and, in order to bring 
the body back again to a state of rest, a quantity of work 
has to be given out just equal to that required to put it in 
motion. This results from the nature of inertia. A body 
in motion may, therefore, be regarded as the representation 
of a quantity of work which can be reproduced upon any 
resistance opposed to its motion. Whilst the body is in 
motion, the work is said to be accumulated. In any given 
instance, the accumulated work acppnds, first, upon the 
mass in motion ; and, secondly, up ^r the velocity with which 
it moves. 

Take the case of a body y jjected vertically upwards in 
vacuum. The projecting £,r<se expends upon the body a 
quantity of work sufficient to raise it through a height equal 
10* 



'J26 MECHANICS. 

to that due to the velocity of projection. Denoting the 
weight of the body, by w, the height to which it rises, by A, 
tnd the accumulated work, by Q, we shall have, 

Q — wh. 

v 2 
But, h = J-, (Art. 116), hence, 



Denoting the mass of the body by m, we shall have, 

w 
m = — (Art. 11), and, by substitution, we have, finally, 

y 

Q = ±mv* ( 136.) 

If the body descends by its own weight, it will have 
impressed upon it by the force of gravity, during the 
descent, exactly the same quantity of work as it gave out 
In ascending. 

The amount of work accumulated in. a body is evidently 
the same, whatever may have been the circumstances under 
which the velocity has been acquired ; and also, the amount 
of work which it is capable of giving out in overcoming any 
resistance is the same, whatever may be the nature of that 
resistance. Hence, the measure of the accumulated work 
of a moving mass is one-half of the mass into the square 
of the velocity. 

The expression my 2 , is called the living force of the 
body. Hence, the living force of a body is equal to its 
mass, multiplied by the square of its velocity. The living 
force of a body is the measure of twice the quantity of 
work expended in producing the velocity, or, it is the 
measure of twice the quantity of work which the body is 
capable of giving out. 

When the forces exerted tend to increase the velocity, 



CURVILINEAR AND ROTARY MOTION. 227 

their work is regarded as positive ; when they tend to dimin- 
ish it, their work is regarded as negative. It is the aggre- 
gate of all the work expended, both positive and negative, 
that is measured by the quantity, ^my*. 

If, at any instant, a body whose mass is m, has a velocity 
v, and, at any subsequent instant, its velocity has become v', 
we shall have, for the accumulated work at these two 
instants, 

Q = \mv\ Q' = imv" ; 

and, for the aggregate quantity of work expended in the 
interval, 

Q" = ±m(v'* - v 2 ) . . . . (137.) 

When the motive forces, during the interval, perform a 
greater quantity of work than the resistances, the value of 
v' will be greater than that of v, and there will be an accu- 
mulation of work in the interval. When the work of the 
resistances exceeds that of the motive forces, the value of v 
will exceed that of v\ Q" will be negative, and there will 
be a loss of living force, which is absorbed by the resistances. 

Living Force of Revolving Bodies. 

149. Denote the angular velocity of a body which is 
restrained by an axis, by 6 ; denote the masses of its ele- 
mentary particles by m, m', &c, and their distances from 
the axis of rotation, by r, r\ &c. Their velocities will be 
rJ, r'&, &c.,. and their living forces will be mr 2 ^ 2 , mV 2 J 2 , &c. 
Denoting the entire living force of the body, by X, we shall 
have, by summation, and recollecting that 8* is the same for 
all the terms, 

L = Z(mry> .... (138.) 

But 2(mr 2 ) is the expression for the moment of inertia of 
the body, taken with respect to the axis of rotation. De- 



228 MECHANICS. 

noting the entire mass by M, its radius of gyration, with 
respect to the axis of rotation, by k, we shall have, 

L = MkH\ 

If, at any subsequent instant, the angular velocity a&s 
become $', we shall, at that instant, have, 

L' = MM* ; 

and, for the loss or gain of living force in the interval, we 
shall have, 

L" = M & 2 (d' 2 .— d 2 ) . . . (139.) 

If we make &' 2 — d 2 = 1, we shall have, 

U" = Mie. = 2(mr 2 ) . . (140.) 

which shows that the moment of inertia of a body, with 
respect to an axis, is equal to the living force lost or 
gained whilst the body is experiencing a change in the 
square of its angular velocity equal to 1. 

The principle of living forces is extensively applied in 
discussing the circumstances of motion of machines. When 
the motive power performs a quantity of work greater than 
that necessary to overcome the resistances, the velocities of 
the parts become accelerated, a quantity of work is stored 
up, to be again given out when the resistances offered 
require a greater quantity of work to overcome them than 
is furnished by the motor. 

In many machines, pieces are expressly introduced to 
equalize the motion, and this is particularly the case when 
either the motive power or the resistance to be overcome, 
is, in its nature, variable. Such pieces are called fly-wheels. 

Fly-Wheels. 

150. A fly-wheel is a heavy wheel, usually of iron, 
mounted upon an axis, near the point of application of the 




CURVILINEAR AND ROTARY MOTION. 229 

force which it is destined to regulate. It is generally com- 
posed of a heavy rim, connected with 
the axis by means of radial arms. 
Sometimes it consists of radiating 
bars, carrying heavy spheres of metal 
at their outer extremity. In either 
case, we see, from Equation 139, that, 
for a given quantity of work absorbed, 
the value of 0' 2 — <3 2 will be less as M 
and k are greater ; that is, the change Fig~i2». 

of angular velocity will be less, as the 
mass of the fly-wheel and its radius of gyration increase. 
It is for this reason that the peculiar form of fly-wheel 
indicated above, is adopted, it being the form that most 
nearly realizes the conditions pointed out. The principal 
objection to large fly-wheels in machinery, is the great 
amount of hurtful resistance which they create, such as fric- 
tion on the axle, &c. Thus, a fly-wheel of 42000 lbs. would 
create a force of friction of 4200 lbs., the coefficient of fric- 
tion being but T \j ; and, if the diameter of the axle were 
8 inches, and the number of revolutions 30 per minute, this 
resistance alone would be equal to 8 horse powers. 

EXAMPLES. 

1. The weight of the ram of a pile-driver is 400 lbs., and 
it strikes the head of a pile with a velocity of 20 feet. 
What is the amount of work stored up in it ? 

SOLUTION. 

The height due to the velocity, 20 feet, is equal to 

— — = 6.22 ft., nearly. 
64i J 

Hence, the stored up work is equal to 

400 lbs. x 6.22 ft. = 2488 lbs. ft. ; 



230 MECHANICS. 

or, the stored up work, equal to half the living force, is 
equal to 

— - x - — — = 2488 units. Ans. 
32i 2 

2. A train, weighing 60 tons, has a velocity of 40 miles 
per hour when the steam is shut off. How far will it travel, 
if no brake be applied, before the velocity is reduced to 10 
miles per hour, the resistance to motion being estimated at 
10 lbs. per ton. Ans. 1 1236 ft. 

Composition of Rotations. 

151. Let a body A CBD, that is free to move, be acted 
upon by a force which, of itself, 

would cause the body to revolve ^ -J& 

for the infinitely small time dt, / / \^ 

about the line AB, with an angu- f g/ ~~^?k\ 

lar velocity v; and at the same I T^p/ \ 

instant, let the body be acted A J}^ 1 C h 

upon by a second force, which \/ J 

would of itself cause the body to ^v. ^/ 

revolve about CD, for the time Fi 130 

dt, with an angular velocity v'. 

Suppose the axes to intersect each other at O, and let P be 
any point in the plane of the axes. Draw PF and PG res- 
pectively perpendicular to OB and C, denoting the for- 
mer, by «, and the latter, by y. Then will the velocity of 
P due to the first force, be equal to vx, and its velocity due 
to the second force will be equal to v'y. Suppose the rota- 
tion to take place in such a manner, that the tendency of 
the rotation about one of the axes, shall be to depress the 
point below the plane, whilst that about the other is to 
elevate it above the plane ; then will the effective velocity 
of P be equal to vx — v'y. If this effective velocity is 0, 
the point P will remain at rest. Placing the expression 
just deduced equal to 0, and transposing, we have, 

vx — v'y. 



CURVILINEAR AND ROTARY MOTION. 231 

To determine the position of P, lay off Oil equal to v, 
01 equal to v', and regard these lines as the representatives 
of two forces ; we have, from the equation, the moment of 
v, with respect to the point P, equal to the moment of v', 
with respect to the same point. Hence, the point P must 
be somewhere upon the diagonal K, of the parallelogram 
described on v, and v'. But P may be anywhere on this 
line ; hence, every point of the diagonal OK, remains at 
rest during the time dt, and is, consequently, the resultant 
axis of rotation. We have, therefore, the following principles : 

If a body be acted upon simultaneously by two forces, 
each tending to impart a motion of rotation about a sepa- 
rate axis, the resultant motion will be one of rotation about 
a third axis lying in the plane of the other two, and passing 
through their common point of intersection. 

The direction of the resultant axis coincides with the 
diagonal of a parallelogram, whose adjacent sides are the 
component axes, and whose lengths are proportional to the 
impressed angular velocities. 

Let OH and 01 represent, as before, the angular veloci- 
ties v and v', and OAT the diagonal of the 
parallelogram constructed on these lines I TC 

as sides. Take any point _ZJ on the second /fX^s' 

axis, and let fall a perpendicular on 077" and /^\ / 
OK; denote the former by r, and the H 

latter, by r" ; denote, also, the resultant Fi s- 18L 

angular velocity, by v". Since the actual space passed over 
by I, during the time t, depends only upon the first force, it 
will be the same whether we regard the revolution as taking 
place about the axis OH, or about the axis OK. If we 
suppose the rotation to take place about OH, the space 
passed over in the time dt, will be equal to rvdt ; if we sup- 
pose the rotation to take place about OK, the space passed 
over in the same time will be equal to r"v"dt. Placing 
these expressions equal to each other, we have, after reduc- 
tion. 

n" = r —v. 



232 MECHANICS. 

But regarding I as a centre of moments, we shall have, 
from the principle of moments, 

OK x r" = vr ; or, OK = -„ v. 

v 

By comparing the last two equations, we have, 

v" = OK. 

That is, the resultant angular velocity will be equal to the 
diagonal of the parallelogram described on the component 
angular velocities as sides. 

By a course of reasoning entirely similar to that employed 
in demonstrating the parallelopipedon of forces, we might 
show, that, 

If a body be acted upon by three simultaneous forces, 
each tending to produce rotation about separate axes inter- 
secting each other, the resultant motion will be one of rota- 
tion about the diagonal of the parallelopipedon whose adja- 
cent edges are the component angular velocities, and the 
resultant angular velocity will be represented by the length 
of this diago?ial. 

The principles just deduced are called, respectively, the 
parallelogram and the parallelopipedon of rotations. 

Application to the Gyroscope. 

152. The gyroscope is an instrument used to illustrate 
the laws of rotary motion. It consists essentially of a heavy 
wheel A, mounted upon 
an axle EC. This axle 
is attached, by means of 
pivots, to the inner edge 
of a circular hoop DE, 
within which the wheel 
A can turn freely. On 

one side of the hoop, and in the prolongation of the axle 
B C, is a bar EE, having a conical hole drilled on its lower 




CURVILINEAR and rotary motion. 233 

face to receive the pointed summit of a vertical standard G. 
It* a string be wrapped several times around the axle B C, 
and then rapidly unwound, so as to impart a rapid motion 
of rotation to the wheel A, in the direction indicated by 
the arrow-head, it is observed that the machine, instead of 
sinking downwards under the action of gravity, takes up a 
retrograde orbital motion about the pivot G, as indicated by 
the arrow-head H. For a time, the orbital motion in- 
creases, and, under certain circumstances, the bar EF is 
observed to rise upwards in a retrograde spiral direction; 
and, if the cavity for receiving the pivot is pretty shallow, 
the bar may even be thrown off the vertical standard. 
Instead of a bar EF, the instrument may simply have an 
ear at E, and be suspended from a point above by means of 
a string attached to the ear. The phenomena observed are 
the same as before. 

Before explaining these phenomena, it will be necessary 
to point out the conventional rules for attributing proper 
signs to the different rotations. 

Let OX, OY, and OZ, be three rectangular axes. It 
has been agreed to call all dis- 
tances, estimated from 0, to- 
wards either _5T, Y, or Z, posi- 
tive / consequently, all distances 
estimated in a contrary direction 
must be regarded as negative. 
If a body revolve about either rig. 133. 

axis, or about any line through 

the origin, in such a manner as to appear to an eye beyond 
it, in the axis and looking towards the origin, to move in 
the same direction as the hands of a watch, that rotation is 
considered positive. If rotation takes place in an opposite 
direction, it is negative. The arrow-head A, indicates the 
direction of positive rotation about the axis of X. To an 
eye situated beyond the body, as at JC, and looking towards 
the origin, the motion appears to be in the same direction 
as the motion of the hands of a watch. The arrowhead B, 



z 

nO } 






' Hi 



234 MECHANICS. 

indicates the direction of positive rotation about the axis 
of T, and the arrow-head (7, the direction of positive rota- 
tion about the axis of Z. 

Suppose the axis of the wheel of the gyroscope to coincide 
with the axis of X, taken horizontal ; let the standard be 
taken to coincide with the axis of Z, the axis of Y~ being 
perpendicular to them both. Let a positive rotation be 
communicated to the wheel by means of a string. For a 
very short time dt, the angular velocity may be regarded 
as constant. In the same time dt, the force of gravity acts 
to impart a motion of positive rotation to the whole instru- 
ment about the axis of Y, which may, for an instant, be 
regarded as constant. Denote the former angular velocity 
by v, and the latter by v r . Lay off in a positive direction 
on the axis of _5T, the distance OD equal to v, and, on the 
positive direction of the axis of Y, the distance OP equal 
to v\ and complete the parallelogram OF. Then (Art. 151) 
will OF represent the direction of the resultant axis of revo- 
lution, and the distance OF will represent the resultant 
angular velocity, which denote by v". In moving from OD 
to OF, the axis takes up a positive, or retrograde orbital 
motion about the axis of Z. To construct the position of 
the resultant axis for the second instant dt, we must com- 
pound three angular velocities. Lay off on a perpendicular 
to OF and OZ, the angular velocity 00 due to the action 
of gravity during the time dt, and on OZ the angular velo- 
city in the orbit ; construct a parallelopipedon on these 
lines, and draw its diagonal through 0. This diagonal 
will coincide in direction with the resultant axis for the 
second instant, and its length will represent the resultant 
angular velocity (Art. 151). For the next instant, we may 
proceed as before, and so on continually. Since, in each 
case, the diagonal is greater than either edge of the paral- 
lelopipedon, it follows that the angular velocity will contin- 
ually increase, and, were there no hurtful resistances, this 
increase would go on indefinitely. The effect of gravity is 
continually exerted to depress the centre of gravity of the 



CURVILINEAR AND ROTARY MOTION. 235 

instrument, whilst the effect of the orbital rotation is to 
elevate it. When the latter effect prevails, the axis of the 
gyroscope will continually rise ; when the former prevails, 
the gyroscope will continually descend. Whether the one 
or the other of these conditions will be fulfilled, depends 
upon the angular velocity of the wheel of the gyroscope, 
and upon the position of the centre of gravity of the instru- 
ment. Were the instrument counterpoised so that the 
centre of gravity would lie exactly over the pivot, there 
would be no orbital motion, neither would the instrument 
rise or fall. Were the centre of gravity thrown oa the 
opposite side of the pivot from the wheel, the rotation due 
to gravity would be negative, that is, the orbital motion 
would be direct \ instead of retrograde. 



236 MECHANICS 



CHAPTER YII. 

MECHANICS OF LIQUIDS. 

Classification of Fluids. 

153. .A fluid is a body whose particles move freely 
amongst, each other, each particle yielding to the slightest 
force. Fluids are of two classes : liquids, of which water is 
a type, and gases, or vapors, of which air and steam are 
types. The distinctive property of the first class is, that 
they are sensibly incompressible/ thus, water, on being 
pressed by a force of 15 lbs. on each square inch of surface, 
only suffers a diminution of about yooV o o" °f ^ ts bulk. The 
second class comprises those which are readily compressible ; 
thus, air and steam are easily compressed into smaller vol- 
umes, and when the pressure is removed, they expand, so as 
to occupy larger volumes. 

Most liquids are imperfect ; that is, there is more or less 
adherence between their particles, giving rise to viscosity. 
In what follows, they will be regarded as destitute of vis- 
cosity, and homogeneous. For certain purposes, fluids may 
also be regarded as destitute of weight, without impairing 
the validity of the conclusions. 

Principle of Equal Pressures. 

154. From the nature and constitution of a fluid, it fol- 
lows, that each of its particles is perfectly movable in all 
directions. From this fact, we deduce the following funda- 
mental law, viz. : If a fluid is in equilibrium under the 
action of any forces whatever, each particle of the mass is 
equally pressed in all directions / for, if any particle were 
more strongly pressed in one direction thao in the others, 




MECHANICS OF LIQUIDS. 237 

it would yield in that direction, and motion \t ould ensue, 
which is contrary to the hypothesis. 

This is called the principle of equal pressures. 

It follows, from the principle of equal pressures, that if 
any point of a fluid in equilibrium, be pressed by any force, 
that pressure will be transmitted without change of intensity 
to every other point of the fluid mass. 

This may be illustrated experimentally, as follows : 

Let AB represent a vessel filled with a fluid in equili- 
brium. Let C and D represent two 
openings, furnished with tightly-fit- 
ting pistons. Suppose that forces are 
applied to the pistons just sufficient to 
maintain the fluid mass in equilibrium. 
If, now, any additional force be appli- 
ed to the piston P, the piston Q will 
be forced outwards ; and in order to 
prevent this, and restore the equili- 
brium, it will be found necessary to apply a force to the 
piston Q, which shall have the same ratio to the force ap- 
plied at P that the area of the piston Q has to the area of 
the piston P. This principle will be found to hold true, 
whatever may be the sizes of the two pistons, or in what- 
ever portions of the surface they may be inserted. If the 
area of P be taken as a unit, then will the pressure upon Q 
be equal to the pressure on P, multiplied by the area of Q. 

The pressure transmitted through a fluid in equilibrium, 
to the surface of the containing vessel, is normal to that sur- 
face ; for if it were not, we might resolve it into two compo- 
nents, one normal to the surface, and the other tangential ; 
the effect of the former would be destroyed by the resistance 
of the vessel, whilst the latter would impart motion to the 
fluid, which is contrary to the supposition of equilibrium. 

In like manner, it may be shown, that the resultant of al 1 
the pressures, acting at any point of the free surface of a 
fluid, is normal to the surface at that point. When the only 
force acting is the force of gravity, the surface is level. For 



238 MECHANICS. 

small areas, a level surface coincides sensibly with a horizon, 
tal plane. For larger areas, as lakes and oceans, a level sur- 
face coincides with the general surface of the earth. Were 
the earth at rest, the level surface of lakes and oceans would 
be spherical ; but, on account of the centrifugal force aris- 
ing from the rotation of the earth, it is sensibly an ellip- 
soidal surface, whose axis of revolution is the axis of the 
earth. 

Pressure due to Weight. 

155. If an incompressible fluid be in a state of equili- 
brium, the pressure at any point of the mass arising from 
the weight of the fluid, is proportional to the depth of the 
point below the free surface. 

Take an infinitely small surface, supposed horizontal, and 
conceive it to be the base of a vertical prism whose altitude 
is equal to its distance below the free surface. Conceive 
this filament to be divided by horizontal planes into infi- 
nitely small, or elementary prisms. It is evident, from the 
principle of equal pressures, that the pressure upon the 
lower face of any one of these elementary prisms is greater 
than that upon its upper face, by the weight of the element, 
whilst the lateral pressures are such as to counteract each 
other's effects. The pressure upon the lower face of the 
first prism, counting from the top, is, then, just equal to its 
weight ; that upon the lower face of the second is equal to 
the weight of the first, plus the weight of the second, and 
so on to the bottom. Hence, the pressure upon the assumed 
surface is equal to the weight of the entire column of fluid 
above it. Had the assumed elementary surface been oblique 
to the horizon, or perpendicular to it, and at the same depth 
as before, the pressure upon it would have been the same, 
from the principle of equal pressures. We have, therefore, 
the following law : 

The pressure upon any elementary portion of the surface 
of a vessel containing a heavy fluid is equal to the weight 
of a prism of the fluid whose base is equal to that surface, 




MECHANICS OF LIQUIDS. 239 

and whose altitude is equal to its depth below the free 
surface. 

Denoting the area of the elementary surface, by s, its 
depth below the free surface, by z, the weight of a unit of 
the Yolunie of the fluid, by w, and the pressure, by p, we 
shall have, 

p = wzs ( 141.) 

We have seen that the pressure upon any element of a 
surface is normal to the surface. Denote 
the angle which this normal makes with 
the vertical, estimated from above, down- 
wards, by (p, and resolve the pressure into 
two components, one vertical and the 
other horizontal, denoting the vertical Ficr 135 

component by p', we shall have, 

p' = wzscosy ( 142.) 

But scoscp is equal to the horizontal projection of the 
elementary surface s, or, in other words, it is equal to a 
horizontal section of a vertical prism, of which that surface 
is the base. Hence, the vertical component of the pressure 
on any element of the surface is equal to the weight of a 
column of the fluid, whose base is equal to the horizontal 
projection of the element, and whose altitude is equal to 
the distance of the element from the upper surface of the 
fluid. 

The distance z has been estimated as positive from the 
surface of the fluid downwards. If 9 < 90°, we have cosp 
positive ; hence, p' will be positive, which shows that the 
vertical pressure is exerted downwards. If <p > 90°, we 
have costp negative / hence, p' is negative, which shows that 
the vertical pressure is exerted upwards (see Fig. 135). 

Suppose the interior surface of a vessel containing a heavy 
fluid to be divided into elementary portions, whose areas 
are denoted by s, s', s", &c. ; denote the distances of these 



240 MECHANICS. 

elements below the upper surface, by s, z\ z'\ &c. From 
the principle just demonstrated, the pressures upon these 
surfaces will be denoted by wsz, ws'z', ws"z", &c, and the 
entire pressure upon the interior of the vessel will be 
equal to, 

w(sz -h s'z' -f- s"z" + &c.) ; or, w x 2(sz). 

Let Z denote the depth of a column of the fluid, whose 
base is equal to the entire surface pressed, and whose weight 
is equal to the entire pressure, then will this pressure be 
equal to w(s + s' -{- s" -\- &e.)Z\ or, ivZ.Zs. Equating 
these values, Ave have, 

w.z(sz)=wZ.Z(s), .: Z = ^ . (143.) 

The second member of (143), (Art. 51), expresses the 
distance of the centre of gravity of the surface pressed, 
below the free surface of the fluid. Hence, 

The entire pressure of a heavy fluid upon the interior of 
the containing vessel, is equal to the weight of a volume of 
the fluid, whose base is equal to the area of the surface 
pressed, and whose altitude is equal to the distance of the 
centre of gravity of the surface from the free surface of the 
fluid. 

EXAMPLES. 

1. A hollow sphere is filled with a liquid. How does the 
entire pressure, on the interior surface, compare with the 
weight of the liquid ? 

SOLUTION. 

Denote the radius of the interior surface of the sphere, 
by r, and the weight of a unit of volume of the liquid, by 
w. The entire surface pressed is measured by 4<r 2 ; and, 
since the centre of gravity of the surface pressed is at a 
distance r below the surface of the liquid, the entire prea- 



MECHANICS OF LIQUIDS. 241 

sure on tne interior surface will be measured by the 
expression, 

w X 47rr 3 X r = 4irwr*. 

But the weight of the liquid is equal to 



twr* 



Hence, the entire pressure is equal to three times the 
weight of the liquid. 

2. A hollow cylinder, with a circular base, is filled with a 
liquid. How does the pressure on the interior surface com- 
pare with the weight of the liquid? 

SOLUTION. 

Denote the radius of the base of the cylinder, by r, and 
the altitude, by h. The centre of gravity of the lateral 
surface is at a distance below the upper surface of the fluid 
equal to \h. If we denote the weight of the unit of volume 
of the liquid, by ic, we shall have, for the entire pressure on 
the interior surface, 

whirr 2 -f- 2w*r . \h 2 = wirrh{r -f- h). 

But the weight of the liquid is equal to 

wid^h. 

r + h 
Hence, the total pressure is equal to times the 

weight of the liquid. 

If we suppose A = r, the pressure will be twice the 
weight. 

If we suppose r — 2A, we shall have the pressure equal 
to \ of the weight. 

If we suppose h = 2r, the pressure will be equal to three 
times the weight, and so on. 
H 



24:2 MECHANICS. 

In all cases, the total pressure will exceed the weight of 
the liquid. 

3. A right cone, with a circular base, stands on its base, 
and is filled with a liquid. How does the pressure on the 
internal surface compare with the weight of the liquid ? 

solution. 

Denote the radius of the base, by r, and the altitude, by 
A, then will the slant height be equal to 



-/A 2 + r\ 

The centre of gravity of the lateral surface, below the 
upper surface of the liquid is equal to §A. If we denote 
the weight of a unit of volume of the liquid, by w, we shall 
have, for the total pressure on the interior surface, 



w*r*h + %wxrh^h* + r* == wtrh{r + \y/h* + r 2 ). 
But the weight of the liquid is equal to 
±wi(i*h = wtrh x \r. 



Sr + 2-v/A 2 + r a 
Hence, the total pressure is equal to ■ 



times the weight. 

4. Required the relation between the pressure and the 
weight in the preceding case, when the cone stands on its 
vertex. 

SOLUTION. 

The total pressure is equal to 



^wxrhyti* -{• r 2 ; 

and, consequently, the pressure is equal to times 

the weight of the liquid 



MECHANICS OF LIQUIDS. 243 

5. What is the pressure on the lateral faces of a cubical 
vessel filled with water, the edges of the cube being 4 feet, 
and the weight of the water 62J lbs. per cubic foot ? 

Ans. 8000 lbs. 

6. A cylindrical vessel is filled with water. The height 
of the vessel is 4 feet, and the radius of the base 6 feet. 
What is the pressure on the lateral surface ? 

Ans. 18850 lbs., nearly. 

Centre of Pressure on a Plane Surface. 

156. Let ABGD represent a plane, pressed by a fluid 
on its upper surface, AB its intersec- 
tion with the free surface of the fluid, ^ A 

G its centre of gravity, the centre i _y2np^. 

of pressure, and s the area of any *^v/ '•% / 

element of the surface at S. De- /^o / 

note the inclination of the plane to 4> / 

the level surface, by a, the perpendic- ^^c 

ular distances from to AB, by x, Fig. 186. 

from G to AB, by p, and from S to 

AB, by r. Denote, also, the entire area A C, by A, and 
the weight of a unit of volume of the fluid, by w. The 
perpendicular distance from G to the free surface of the 
fluid, will be equal to p sina, and that of any element of the 
surface, will be r sina. 

From the preceding article, it follows that the entire 
pressure exerted is equal to wAp sina, and its moment, with 
respect to AB as an axis of moments, is equal to 

wAp sina x x. 

The elementary pressure on s is, in like manner, equal to 
wsr sina, and its moment, with respect to AB, is wsr* sina, 
and the sum of all the elementary moments is equal to 

w sina 2(sr 2 ). 



24-i MECHANICS. 

But the resultant moment is equal to the algebraic sum 
of the elementary moments. Hence, 

wAp sina x x = w sina 2(sr*) ; 

and, by reduction, 

■■-3? <■«-> 

The numerator is the moment of inertia of the plane 
ABCD, with respect to AJB, and the denominator is the 
moment of the area with respect to the same line. Hence, 
the distance from the centre of pressure to the intersection 
of the plane with the free surface, is equal to the moment 
of inertia of the plane, divided by the moment of the 
plane. 

If we take the straight line AD, perpendicular to AB, as 
an axis of moments, denoting the distance of from it, by 
y, and of s from it, by /, we shall, in a similar manner, have, 

wAp sinay = wsmoi2(srl) ; 

and, by reduction, 

* = %* < 145 -> 

The values of x and y make known the position of the 
centre of pressure. 

EXAMPLES . 

1. "What is the position of the centre of pressure on a 
rectangular flood-gate, the upper line of the gate coinciding 
with the surface of the water ? 

SOLUTION. 

It is obvious that it will be somewhere on the line joining 
the middle points of the upper and lower edges of the gate. 



MECHANICS OF LIQUIDS. 245 

Denote its distance from the upper edge, by s, the depth of 
the gate, by 21, and its mass, by 31. The distance of the 
centre of gravity from the upper edge will be equal to I. 

From Example 1 (Art. 132), replacing d by I, and 
reducing, we have, for the moment of inertia of the 
rectangle, 

M(^ + F)= Mil'. 

But the moment of the rectangle is equal to, 

Ml; 

hence, by division, we have, 

Z = i* = §(20. 

That is, the centre of pressure is at two-thirds of the 
distance from the upper to the lower edge of the gate. 

2. Let it be required to find the pressure on a submerged 
rectangular flood-gate ABC J), the plane of 
the gate being vertical. Also, the distance J G Y 

of the centre of pressure below the surface 
of the water. 



SOLUTION. 



C* 



» c 

Fig. 137. 



Let EF be the intersection of the plane 
with the surface of the water, and suppose 
the rectangle AG to be prolonged till it 
reaches EF. Let C, G\ and G", be the centres of pressure 
of the rectangles EG, EB, and A G respectively. Denote 
the distance GC", by z, the distance ED, by a, and the 
distance EA, by a'. Denote the breadth of the gate, by b, 
and the weight, a unit of volume of the water, by w. 

The pressure on EC will be equal to \a?bw, and the pres- 
sure on EB will be equal to \a' 2 bw ; hence, the pressure on 
A C will be equal to 

\bw(a? — a' 3 ) ; 

which is the pressure required. 



246 MECHANICS. 

From the principle of moments, the moment of the pres 
sure on A 6 Y , is equal to the moment of the pressure on JEQ 
minus the moment of the pressure on EJB. Hence, from 
the last problem, 

±bw(a 2 — a n ) X z = ±bwa? x f a — \bwa"> x §«', 
9 a 3 - a n 

n — _ A. 



which is the required distance from the surface of the 
water. 

3. Let it be required to find the pressure on a rectangular 
flood-gate, when both sides are pressed, 
the water being at different levels on 'IBHH 
the two sides. Also, to find the centre 
of pressure. 

SOLUTION. 

Denote the depth of water on one F{a , m 

side by a, and on the other side, by 
a\ the other elements being the same as before. 

The total pressure will, as before, be equal to, 

±bw{a* - a' 2 ). 
Estimating z from C upwards, 



a' 



Ans. 



4. A sluice-gate, 10 feet square, is placed vertically, its 
upper edge coinciding with the surface of the water What 
is the pressure on the upper and lower halves of the gate, 
respectively, the weight of a cubic foot of water being 
taken equal to 62| lbs. ? Ans. 7812.5 lbs., and 23437.5 lbs. 

5. What must be the thickness of a rectangular dam of 
granite, that it may neither rotate about its outer angular 



MECHANICS OF LIQUIDS. 247 

point nor slide along its base, the weight of a cubic foot of 
granite being 160 lbs., and the coefficient of friction between 
it and the soil being .6 ? 

SOLUTION. 

First, to find the thickness necessary to prevent rotation 
outwards. Denote the height of the wall, by A, and sup- 
pose the water to extend from the bottom to the top. De- 
note the thickness, by t, and the length of the wall, or dam, 
by I. The weight of the wall in pounds, will be equal to 

Iht X 160 ; 

and this being exerted through its centre of gravity, the 
moment of the weight with respect to the outer edge, as an 
axis, will be equal to 

Itflh X 160 = 80lht\ 



The pressure of the water against the inner face, in 
pounds, is equal to 

\W X62.5 = lh* X 31.25. 

This pressure is applied at the centre of pressure, which 
is (Example 1) at a distance from the bottom of the wall 
equal to \h ; hence, its moment with respect to the outer 
edge of the wall, is equal to 

W x 10.4166. 

The pressure of the water tends to produce rotation out- 
wards, and the weight of the wall acts to prevent this rota- 
tion. In order that these forces may be in equilibrium, 
their moments must be equal ; or 

80lhf = Ih s X 10.4166. 



248 MECHANICS. 

Whence, we find, 



t — Ayj302 = .36 X h. 

Next, to find the thickness necessary to prevent sliding 
along the base. The entire force of friction due to th« 
weight of the wall, is equal to 

IGOlht X .6 = 9Qlht; 

and in order that the wall may not slide, this must be equal 
to the pressure exerted horizontally against the wall. Hence, 

96lht = 3l.25lh*. 

Whence, we find, 

t = .325A. 

If the wall is made thick enough to prevent rotation, it 
will be secure against sliding. 

6. What must be the thickness of a rectangular dam 
15 feet high, the weight of the material being 140 lbs. to 
the cubic foot, that, when the water rises to the top, the 
structure may be just on the point of overturning ? 

Ans. 5.7 ft. 

7. The staves of a cylindrical cistern filled with water, are 
held together by a single hoop. Where must the hoop be 
situated ? 

Ans. At a distance from the bottom equal to one-third of 
the height of the cistern. 

8. Required the pressure of the sea on the cork of an 
empty bottle, when sunk to the depth of 600 feet, the 
diameter of the cork being f of an inch, and a cubic foot cf 
sea water being estimated to weigh 64 lbs. ? Ans. 134 lbs. 



MECHANICS OF LIQUIDS. 249 

Buoyant Effort of Fluids. 

157. Let A represent any solid body suspended in a 
heavy fluid. Conceive this solid to be divided 
into vertical prisms, whose horizontal sections are 
infinitely small. Any one of these prisms will be 
pressed downward by a force equal to the weight ® 

of a column of fluid, whose base (Art. 155) is Fi 139 " 
equal to the horizontal section of the filament, 
and whose altitude is the distance of its upper surface from 
the surface of the fluid ; it will be pressed upward by a 
force equal to the weight of a column of fluid having the 
same base and an altitude equal to the distance of the lower 
base of the filament from the surface of the fluid. The re- 
sultant of these two pressures is a force exerted vertically 
upwards, and is equal to the weight of a column of fluid, 
equal in bulk to that of the filament and having its point 
of application at the centre of gravity of the volume of the 
filament. This being true for each filament of the body, 
and the lateral pressures being such as to destroy each 
other's effects, it follows, that the resultant of all the pres- 
sures upon the body will be a vertical force exerted upwards, 
whose intensity is equal to the weight of a portion of the 
fluid, whose volume is equal to that of the solid, and the 
point of application of which is the centre of gravity of the 
volume of the displaced fluid. This upward pressure is call- 
ed the buoyant effort of the fluid, and its point of application 
is called the centre of buoyancy. The line of direction of 
the buoyant effort, in any position of the body, is called a 
line of support. That line of support which passes through 
the centre of gravity of a body, is called the line of rest. 

Floating Bodies. 

158. A body wholly or partially immersed in a heavy 
fluid, is urged downwards by its weight applied at its cen- 
tre of gravity, and upwards, by the buoyant effort of the 
fluid applied at the centre of buoyancy. 
11* 



250 



MECHANICS. 



M" 



E 



The body can only be in equilibrium when the line through 
the centre of gravity of the body, and the centre of buoy- 
ancy, is vertical ; in other words, when the line of rest is ver- 
tical. When the weight of the body exceeds the buoyant 
effort, the body will sink to the bottom ; when they are 
just equal, it will remain in equilibrium, wherever placed hi 
the fluid. When the buoyant effort is greater than the 
weight, it will rise to the surface, and after a few oscillations, 
?\ T ill come to a state of rest, in such a position, that the 
weight of the displaced fluid is equal to that of the body, 
when it is said to float. The upper surface of the fluid is 
then called the plane of floatation,, and its intersection with 
the surface of the body, the line of floatation. 

If a floating body be slightly disturbed from its position 
of equilibrium, the centres of grav- 
ity and buoyancy will no longer 
be in the same vertical line. Let 
DE represent the plane of floata- 
tion, G the centre of gravity of the 
body (Fig. 141), GH its line of rest, 
and G the centre of buoyancy in 
the disturbed position of the 
body. 

If the line of support GB, in- 
tersects the line of rest in M, 
above G, as in Fig. 141, the buoy- 
ant effort and the weight will conspire to restore the body 
to its position of equilibrium ; in this case, the equilibrium 
must be stable. 

If the point M falls below G, 
as in Fig. 142, the buoyant ef- 
fort and the weight will conspire 
to overturn the body ; in this 
case, the body must, before be- 
ing disturbed, have been in a 
state of unstable equilibrium. 

If the centre of buoyancy and centre of gravity are 



Fig. 140. 






MECHANICS OF LIQUIDS. 251 

always on the same vertical, the point 

M will coincide with G (Fig. 143), 

and the body will be in a state of 

indifferent equilibrium. The limiting 

position of the point M, or of the 

intersection of the lines of rest and Fi 148 

of support, obtained by disturbing the 

floating body through an infinitely small angle, is called the 

'metacentre of the body. Hence, 

If the metacentre is above the centre of gravity of the 
body, it will be in a state of stable equilibrium,, the line of 
rest being vertical; if it is below the centre of gravity, the 
body will be in unstable equilibrium / if the two points 
coincide, the body loill be in indifferent equilibrium. 

The stability of the floating body will be the greater, as 
the metacentre is higher above the centre of gravity. This 
condition is practically fulfilled in loading ships, or other 
floating bodies, by stowing the heavier objects nearest the 
bottom of the vessel. 

Specific Gravity. 

159. The specific gravity of a body is its relative weight ; 
that is, it is the number of times the body is heavier than 
an equivalent volume of some other body taken as a 
standard. 

The numerical value of the specific gravity of any body, 
is the quotient obtained by dividing the weight of any 
volume of the body by that of an equivalent volume of the 
standard. 

For solids and liquids, water is generally taken as the 
6tandard, and, since this liquid is of different densities at 
different temperatures, it becomes necessary to assume also 
a standard temperature. Most writers have taken 60° 
Fahrenheit as this standard. Some, however, have taken 
3 8° 75 Fah., for the reason that experiment has shown that 
water has its maximum density at this temperature. We 
shall adopt the latter standard, remarking that specific 



252 MECHANICS. 

gravities, determined at any temperature, may be readily 
reduced to what they would have been had they been deter- 
mined at any other temperature. 

The densities of pure water at different temperatures has 
been determined with great accuracy by experiment, and 
the results arranged in tables, the density at 3 8° 75 being 
taken as 1. 

Since the specific gravity of a body increases as the 
density of the standard diminishes, it will be a little less 
when referred to water at 3 8° 75 than at any other tempe- 
rature. 

Let d and d' denote the densities of water at any two 
temperatures t and t' ; let s and s' denote the specific 
gravities of the same body, referred to water at these 
temperatures; then, 

s : s' : : d' : d, /. s = ~ . ( 146.) 

This formula is applicable in any case where it is necessary 
to reduce the specific m gravity taken at the temperature f 
to what it would have been if taken at the temperature t. 
If t = 38°75, we have d = 1, and the formula becomes, 

s = s'd f (147.) 

Hence, to reduce the specific gravity taken at the tem- 
perature t\ to the standard temperature, multiply it by 
the tabular density of water at the temperature t'. 

The specific gravity should also be corrected for expan- 
sion. This correction is made in a manner entirely similar 
to the last. Denote the volumes of the same body at the 
temperatures t and t\ by v and v\ and the apparent specific 
gravities, after the last correction, by S and S', then, 

S: & ;':Vi t>, .\ £.= — (148.) 



MECHANICS OF LIQUIDS. 253 

If t is the standard temperature, and v the unit of volume 
we have, 

8 = '■& X v' . . . . ( 149.) 

In what follows, we shall suppose that the specific gravi- 
ties are taken at the standard temperature, in which case 
no correction will be necessary. 

Gases are generally referred to atmospheric air as a 
standard, but, as air may be readily referred to water as a 
standard, we shall, for the purpose of simplification, suppose 
that the standard for all bodies is distilled water at 3 8° 75 
Fahrenheit. 

Hydrostatic Balance. 

160. This balance is similar to 
that described in Article 81, ex- . 

cept the scale-pans have hooks at- 
tached to their lower surfaces for 
the purpose of suspending bodies. 
The suspension is effected by a 
fine platinum wire, or by some 



m 



other material not acted upon by Fig. 144. 

the liquids employed. 

To determine the Specific Gravity of an Insoluble Body. 

161. Attach the suspending wire to the first scale-pan, 
and after allowing it to sink in a vessel of water to a certain 
depth, counterpoise it by an equal weight, attached to the 
hook of the second scale-pan. Place the body in the first 
scale-pan, and counterpoise it by weights in the second pan. 
These weights will give the weight of the body in air. 
Next, attach the body to the suspending wire, and immerse 
it in the water. The buoyant effort of the water will be 
equal to the weight of a volume of water equivalent to that 
ol the body (Art. 157) ; hence, the second pan will descend. 
Restore the equilibrium by weights placed in the first pan. 
These weights will give the weight of the displaced water 



254 MECHANICS. 

Divide the weight of the body in air by the weight just 
found, and the quotient will be the specific gravity sought. 
If the body will not sink in water, determine its weight in 
air as before ; then attach to it a body so heavy, that the 
combination will sink ; find, as before, the loss of weight of 
the combination, and also the loss of weight of the heavier 
body ; take the latter from the former, and the difference 
will be the loss of weight of the lighter body ; divide its 
weight in air by this weight, and the quotient will be the 
specific gravity sought. 

If great accuracy is required, account must be taken of 
the buoyant effort of the air, which, when the body is very 
light, and of considerable dimensions, will render the appa- 
rent weight less than the true wxight, or the weight in 
vacuum. Since the weights used in counterpoising are 
always very dense, and of small dimensions, the buoyant 
effort of the air upon them may always be neglected. 

To determine the true weight of a body in vacuum : let 
w denote its weight in air, w' its weight in water, and IV its 
weight in vacuum; then will W — w, and W— w\ denote 
its loss of weight in air and water ; denote the specific 
gravity of air referred to water, by s. Since the losses of 
weight in air and water arc proportional to their specific 
gravities, we have, 

W — w : W — w' : : * : 1 ; or, W — w — sW — sw f , 

__ w — sw f 

W = — • 

1 — s 

This weight should be used, instead of the weight in air. 

To determine the Specific Gravity of Liquids. 

162. First Method. — Take a vial with a narrow neck, 
and weigh it; fill it w r ith the liquid, and weigh again; 
empty out the liquid, and fill with water, and weigh again ; 
deduct from the last two weights, respectively, the weight 
of the vial ; these results will give the weights of equa! 



MECHANICS OF LIQUIDS. 255 

Volumes of the liquid and of water. Divide the former by 
the latter, and the quotient will be the specific gravity 
sought. 

Second Method. — Take a heavy body, that will sink both 
in the liquid and in water, and which will not be acted upon 
by either ; determine its loss of weight, as already explained, 
first in the liquid, then in water ; divide the former by the 
latter, and the quotient will be the specific gravity sought. 
The reason is evident. 

Third Method. — Let AB and CD represent two 
graduated glass tubes of half an inch in 
diameter, open at both ends. Let their /^^ 

upper ends communicate with the receiver [ ^ ^ | 

of an air-pump, and their lower ends dip — 
into two cisterns, one containing distilled 
water, and the other the liquid whose AP fl i ilc 

specific gravity is to be determined. Let Fi 145 

the air be partially exhausted from the 
receiver by means of an air-pump ; the liquids will rise in 
the tubes, but to different heights, these being inversely as 
the specific gravities of the liquids. If we divide the height 
of the column of water by that of the other liquid, the 
quotient will be the specific gravity sought. By creating 
different degrees of rarefaction, the columns will rise to 
different heights, but their ratios ought to be the same. We 
are thus enabled to make a series of observations, each cor- 
responding to a different degree of rarefaction, from which 
a more accurate result can be had than from a single obser- 
vation. 

To determine the Specific Gravity of a Soluble Body. 

163. Find its specific gravity by the method already 
given, with respect to some liquid in which it is not soluble, 
and find also the specific gravity of this liquid referred to 
water; take the product of these specific gravities, and it 
will be the specific gravity sought. For, if the body is m 
times heavier than an equivalent volume of the liquid used, 



256 MECHANICS. 

and this is n times heavier than an equivalent volume of 
water, it follows that the body is mn times heavier than its 
volume of water, whence the rule. 

The auxiliary liquid, in some cases, might be a saturated solu- 
tion of the given body in water ; the rule remains unchanged. 

To determine the Specific Gravity of the Air. 

164. Take a hollow globe, fitted with a stop-cock, to 
shut off communication with the external air, and, by means 
of the air-pump or condensing syringe, pump in as much air 
as is convenient, close the stop-cock, and weigh the globe 
thus filled. Provide a glass tube, graduated so as to show 
cubic inches and decimals of a cubic 

inch, and, having filled it with mer- 
cury, invert it over a mercury bath. 
Open the stopcock, and allow the com- 
pressed air to escape into the inverted 
tube, taking care to bring the tube 
into such a position that the mercury 
without the tube is at the same level Fig. 146. 

as within. The reading on the tube 
will give the volume of the escaped air. Weigh the globe 
again, and subtract the weight thus found from the first 
weight; this difference will indicate the weight of the 
escaped air. Having reduced the measured volume of air 
to what it would have occupied at a standard temperature 
and barometric pressure, by means of rules yet to be 
deduced, compute the weight of an equivalent volume of 
water ; divide the weight of the corrected volume of air by 
that of an equivalent volume of distilled water, and the 
quotient will be the specific gravity sought. 

To determine the Specific Gravity of a Gas. 

165. Take a glass globe of suitable dimensions, fitted 
with a stop-cock for shutting off communication with the 
atmosphere. Fill the globe with air, and determine the 
weight of the globe thus filled referred to a vacuum, as 
already explained. From the known volume of the globe 




MECHANICS OF LIQUIDS. 257 

and the specific gravity of air, the weight of the contained 
air can be computed; subtract this from the previous 
weight, and we shall have the true weight of the globe 
alone; determine in succession the weights of the globe 
filled with water and with the gas in vacuum, and from each 
subtract the weight of the globe ; divide the latter result by 
the former ; the quotient will be the specific gravity required. 

Hydrometers. 

166. A hydrometer is a floating body, used for the pur- 
pose of determining specific gravities. Its construction de- 
pends upon the principle of floatation. Hydrometers are 
of two kinds. 1. Those in which the submerged volume is 
constant. 2. Those in which the weight of the instrument 
remains constant. 

Nicholson's Hydrometer. 

167. This instrument consists of a hollow brass cylinder 
-4, at the lower extremity of which is fastened 

a basket B, and at the upper extremity a wire, C 

bearing a scale-pan C. At the bottom of the 
basket is a ball of glass E, containing mer- 
cury, the object of which is, to cause the in- 
strument to float in an upright position. By 
means of this ballast, the instrument is ad- 
justed so that a weight of 500 grains, placed 
in the pan 6 r , will sink it in distilled water to 
a notch _Z>, filed in the neck. 



To determine the specific gravity of a solid ' Fig, i 47 . 
which weighs less than 500 grains. Place the 
body in the pan (7, and add weights till the instrument 
sinks, in distilled water, to the notch D. The added 
weights, substracted from 500 grains, will give the weight 
of the body in air. Place the body in the basket B, which 
generally has a reticulated cover, to prevent the body from 
floating away, and add other weights to the pan, until the 
instrument again sinks to the notch D. The weights last 
added give the weight of the water displaced by the body 



258 MECHANICS. 

Divide the first of these weights by the second, and tne 
quotient will be the specific gravity required. 

To find the specific gravity of a liquid. Having carefully 
weighed the instrument, place it in the liquid, and add 
weights to the scale-pan till it sinks to D. The weight of 
the instrument, plus the sum of the weights added, will be 
the weight of the liquid displaced by the instrument. Next, 
place the instrument in distilled water, and add weights till 
it sinks to D. The weight of the instrument, plus the added 
weights, gives the weight of the displaced water. Divide 
the first result by the second, and the quotient will be the 
specific gravity required. The reason for this rule is evident. 

A modification of this instrument, in which the basket JB, 
is omitted, is sometimes constructed for determining specific 
gravities of liquids only. This kind of hydrometer is 
generally made of glass, that it may not be acted upon 
chemically, by the liquids into which it is plunged. The 
hydrometer just described, is generally known as Fahren 
heifs hydrometer, or Fahrenheit's areometer. 

Scale Areometer. 
16§. The scale areometer is a hydrometer whose w r eight 
remains constant ; the specific gravity of a liquid is made 
known by the depth to which it sinks in it. The 
instrument consists of a hollow glass cylinder A, o 

with a stem (7, of uniform diameter. At the 
bottom of the cylinder is a bulb B, containing 
mercury, to make the instrument float upright. 
By introducing a suitable quantity of mercury, 
the instrument may be adjusted so as to float at 
any desired point of the stem. When it is de- 
signed to determine the specific gravities of liquids, 
both heavier and lighter than water, it is bal- 
lasted so that in distilled water, it will sink to the Fig. 143. 
middle of the stem. This point is marked on the 
stem with a file, and since the specific gravity of water is 1, 
it is numbered 1 on the scale. A liquid is then formed by 
dissolving common salt in water whose specific gravity w 



MECHANICS OF LIQUIDS. 259 

1.1, and the instrument is allowed to float freely in it; the 
point E, to which it then sinks, is marked on the stem, and 
the intermediate part of the scale, HE, is divided into 10 
equal parts, and the graduation continued above and below 
throughout the stem. The scale thus constructed is marked 
on a piece of paper placed within the hollow stem. To use this 
hydrometer, we have simply to put it into the liquid and 
allow it to come to rest ; the division of the scale which cor- 
responds to the surface of floatation, makes known the spe- 
cific gravity of the liquid. The hypothesis on which this 
instrument is graduated, is, that the increments of specific 
gravity are proportional to the increments of the submerged 
portion of the stem. This hypothesis is only approximately 
true, but it approaches more nearly to the truth as the dia- 
meter of the stem diminishes. 

When it is only desired to use the instrument for liquids 
heavier than water, the instrument is ballasted so that the 
division 1 shall come near the top of the stem. If it is to 
be used for liquids lighter than water, j£ is ballasted so that 
the division 1 shall fall near the bottom of the stem. In 
this case we determine the point 0.9 by using a mixture of 
alcohol and water, the principle of graduation being the same 
as in the first instance. 

Volumeter. 

169. The volumeter is a modification of the scale areo- 
meter, differing from it only in the method of graduation. 
The graduation is effected as follows : The instru- 
ment is j)laced in distilled water, and allowed to 
come to a state of rest, and the point on the stem 
where the surface cuts it, is marked with a file. 
The submerged volume is then accurately deter- 
mined, and the stem is graduated in such a man- 
ner that each division indicates a volume equal to 
a hundredth part of the volume originally sub- 
merged. The divisions are then numbered from 
the first mark in both directions, as indicated in B 
the figure. To use the instrument, place it in the F,g ' 149# 
iquid, and note the division to which it sinks ; 



J 



260 MECHANICS. 

divide 100 by the number indicated, and the quotient will 
be the specific gravity sought. The principle employed is, 
that the specific gravities of liquids are inversely as the vol- 
umes of equal weights. Suppose that the instrument indi- 
cates x parts ; then the weight of the instrument displaces 
x parts of the liquid, whilst it displaces 100 parts of 
water. Denoting the specific gravity of the liquid by IS, and 
that of water by 1, we have, 

S : 1 :: 100 : x, .\ S = — • 

x 

A table may be computed to save the necessity of per 
forming the division. 

Densimeter. 

170. The densimeter is a modification of the volum- 
eter, and admits of use when only a small portion of the 
liquid can be had, as is often the case in examining 
animal secretions, such as bile, chyle, &c. The 
construction of the densimeter differs from that of 
the volumeter, last described, in having a small 
cup at the upper extremity of the stem, destined 
to receive the fluid whose specific gravity is to be 
determined. 

The instrument is ballasted so that when the cup 
is empty, the densimeter will sink in distilled water 
to a point B, near the bottom of the stem. This 
point is the of the instrument. The cup is then 
filled with distilled water, and the point (7, to rig. 150. 
which it sinks, is marked ; the space B (7, is divi- 
ded into any number of equal parts, say 10, and the grad- 
uation is continued to the top of the tube. 

To use the instrument, place it in distilled water, and fill 
the cup with the liquid iu question, and note the division to 
which it sinks. Divide 10 by the number of this division, 
and the quotient will be the specific gravity required. The 
principle of the densimeter is the same as that of the volu 
meter. 



MECHANICS OF LIQUIDS. 261 

Centesimal Alcoholometer of Gay Lussac. 

171. This instrument is the same in construction as the 
scale areometer ; the graduation is, however, made on a diff- 
erent principle. Its object is, to determine the percentage oi 
alcohol in a mixture of alcohol and water. The graduation is 
made as follows : the instrument is first placed in absolute 
alcohol, and ballasted so that it will sink nearly to the top 
of the stem. This point is marked ] 00. Next, a mixture 
of 95 parts of alcohol and -6 of water, is made, and the point 
to which the instrument sinks, is marked 95. The inter- 
mediate space is divided into 5 equal parts. Next, a mix- 
ture of 90 parts of alcohol and 10 of water is made ; the 
point to which the instrument sinks, is marked 90, and the 
space between this and 95, is divided into 5 equal parts. In 
this manner, the entire stem is graduated by successive 
operations. The spaces on the scale are not equal at differ- 
ent points, but, for a space of five parts, they may be re- 
garded as equal, without sensible error. 

To use the instrument, place it in the mixture of alcohol 
and water, and read the division to which it sinks ; this will 
indicate the percentage of alcohol in the mixture. 

In all of the instruments, the temperature has to be taken 
into account ; this is usually effected by means of correc- 
tions, which are tabulated to accompany the different 
instruments. 

On the principle of the alcoholometer, are constructed a 
great variety of areometers, for the purpose of determining 
the degrees of saturation of wines, syrups, and other liquids 
employed in the arts. # 

In some nicely constructed hydrometers, the mercury 
used as ballast serves also to fill the bulb of a delicate ther- 
mometer, whose stem rises into the cylinder of the instru- 
ment, and thus enables us to note the temperature of the 
fluid in which it is immersed. 

EXAMPLES. 

I. A cubic foot of water weighs 1000 ounces. Required 



262 mechanics. 

the weight of a cubical block of stone, one of whose edges 
is 4 feet, its specific gravity being 2.5. Ans. 10000 lbs. 

2. Required the number of cubic feet in a body whose 
weight is 1000 lbs., its specific gravity being 1.25. 

Ans. 12.8. 

3. Two lumps of metal weigh respectively 3 lbs., and 1 lb., 
and their specific gravities are 5 and 9. What will be the 
specific gravity of an alloy formed by melting them together, 
supposing no contraction of volume to take place. 

Ans. 5.625. 

4. A body weighing 20 grains has a specific gravity of 2.5. 
Required its loss of weight in water. Ans. 8 grains. 

5. A body weighs 25 grains in water, and 40 grains in a 
liquid whose specific gravity is .7. What is the weight of 
the body in vacuum ? Ans. 75 grains. 

6. A Nicholson's hydrometer weighs 250 grains, and it 
requires an additional weight of 336 grains to sink it to the 
notch in the stem, in a mixture of alcohol and water. What 
is the specific gravity of the mixture? Ans. .781. 

7. A block of wood is found to sink in distilled water till 
^ of its volume is submerged. What is its specific gravity ? 

Ans. .875. 

8. The weight of a piece of cork in air, is f oz. ; the 
weight of a piece of lead in water, is 6f oz. ; the weight of 
the cork and lead together in water, is 4 t £q oz. What is 
the specific gravity of the cork ? Ans. 0.24. 

9. A solid, whose weight is 250 grains, weighs in water, 
147 grains, and, in another fluid, 120 grains. What is the 
specific gravity of the latter fluid ? Ans. 1.26°. 

10. A solid weighs 60 grains in air, 40 in water, and 30 in 
an acid. What is the specific gravity of the acid ? 

Ans. 1.5. 



MECHANICS OF LIQUIDS. 



263 



The following table of the specific gravity of some of the 
most important solid and fluid bodies, is compiled from a 
table given in the Ordnance Manual. 

TABLE OF SPECIFIC GRAVITIES OF SOLIDS AND LIQUIDS. 



Antimony, cast 

Brass, cast 

Copper, cast 

Gold, hammered 

Iron, bar 

Iron, cast 

Lead, cast 

Mercury at 32° F 

" at 60° 

Platina, rolled 

" hammered. . . 
Silver, hammered. . . . 

Tin, cast 

Zinc, cast 

Bricks 

Chalk 

Coal, bitumiuous 

Diamond 

Earth, common 

Gypsum 

Ivory 



SPEC. GBAV. 



6.712 

8.396 

8 788 

19.361 

7.788 

7.207 

11.352 

13.598 

13.580 

22.069 

20.337 

10.511 

7.291 

6.861 

1.900 

2.784 

1.270 

3.521 

1 500 

2.168 

1.822 



Limestone 

Marble, common. 
Salt, common . . . 

Sand 

Slate 

Stone, common . . 

Tallow 

Boxwood 

Cedar 

Cherry 

Lignum vita? .... 

Mahogany 

Oak, heart 

Pine, yellow 

Nitric acid 

Sulphuric acid. . . 
Alcohol, absolute . 
Ether, sulphuric . 

Sea water 

Olive oil 

Oil of Turpentine 



SPEC. GHAV. 



3.180 
2.686 
2.130 
1 . 800 
2.672 
2.520 
0.945 
0.912 
0.596 
0.715 
1.333 
0.854 
1.170 
. 660 
1.217 
1.841 
0.792 
0.715 
1.026 
0.915 
0.870 



Thermometer. 

172. A thermometer is an instrument used for measur- 
ing the temperatures of bodies. It is found, by observation, 
that almost all bodies expand when heated, and contract 
when cooled, so that, other things being equal, they always 
occupy the same volumes at the same temperatures. It is 
also found that different bodies expand and contract in a 
different ratio for the same increments of temperature. As 
a general rule, liquids expand much more rapidly than solids, 
and gases much more rapidly than liquids. The construc- 
tion of the thermometer depends upon this principle of 
unequal expansibility of different bodies. A great variety 
of combinations have been used in the construction of the** 



264 MECHANICS. 

mometers, only one of which, the common mercuriai thet 
mometer, will be described. 

The mercurial thermometer consists of a cylindrical or 
spherical bulb A, at the upper extremity of which, 
is a narrow tube of uniform bore, hermetically 
sealed at its upper end. The bulb and tube are 
nearly filled with mercury, and the whole is 
attached to a frame, on which is a scale for deter- 
mining the temperature, which is indicated by the 
rise and fall of the mercury in the tube. 

The tube should be of uniform bore through- 
out, and, when this is the case, it is found that 
the relative exjiansion of the mercury and glass 
is very nearly uniform for constant increments of 
temperature. A thermometer maybe constructed 
and graduated as follows : A tube of uniform ^""ii 
bore is selected, and upon one extremity a bulb is 
blown, which may be cylindrical or spherical ; the former 
shape is, on many accounts, the preferable one. At the 
other extremity, a conical-shaped funnel is blown open at 
the top. The funnel is filled with mercury, which should be 
of the purest quality, and the whole being held vertical, the 
heat of a spirit-lamp is applied to the bulb, which expand- 
ing the air contained in it, forces a portion in bubbles up 
through the mercury in the funnel. The instrument is next 
allowed to cool, when a portion of mercury is forced down 
the capillary tube into the bulb. By a repetition of this 
process, the entire bulb may be filled with mercury, as well 
as the tube itself. Heat is then applied to the bulb, until 
the mercury is made to boil ; and, on being cooled down to 
a little above the highest temperature which it is desired to 
measure, the top of the tube is melted off by means of a 
jet of flame, urged by a blow-pipe, and the whole is her- 
metically sealed. The instrument, thus prepared, is attached 
to a frame, and graduated as follows : 

The instrument is plunged into a bath of melting ice, 
and, after being allowed to remain a sufficient time for the 



MECHANICS OF LIQUIDS. 265 

parts of the instrument to take the uniform temperature of 
the melting ice, the height of the mercury in the tube is 
marked on the scale. This gives the freezing point of the 
scale. The instrument is next plunged into a bath of boiling- 
water, and allowed to remain long enough for all of the parts 
to acquire the temperature of the water and steam. The 
height of the mercury is then marked on 'the scale. This 
gives the boiling point of the scale. The freezing and 
boiling points having been determined, the intermediate 
space is divided into a certain number of equal parts, 
according to the scale adopted, and the graduation is then 
continued, both upwards and downwards, to any desired 
extent. Three principal scales are used. Fahrenheit's 
scale, in which the space between the freezing and boiling 
point is divided into 1 80 equal parts, called degrees, the 
freezing point being marked 32°, and the boiling point 212°. 
In this scale, the point is 32 degrees below the freezing 
point. The Centigrade scale, in which the space between 
the fixed points is divided into 100 equal parts, called 
degrees. The of this scale is at the freezing point. 
Reaumur's scale, in which the same space is divided into 
80 equal parts, called degrees. The of this scale also is 
at the freezing point. 

If we denote the number of degrees on the Fahrenheit, 
Centigrade, and Reaumur scales, by F, C, and K respec- 
tively, the following formula will enable us to pass from 
any one of these scales to any other : 

i(i^°-32) =iC° = \E°. 

The scale most in use in this country is Fahrenheit's 
The other two are much used in Europe, particularly the 
Oeutigrade scale. 

Velocity of a liquid flowing through a small orifice. 

173. Let ABD represent a vessel, having a very small 
orifice at its bottom, and filled with any liquid. 
12 




266 MECHANICS. 

Denote the area of the orifice, by a, and its 
depth below the upper surface, by h. Let D 
represent an infinitely small layer of the liquid 
situated at the orifice, and denote its height, 
by h' . This layer is (Art. 155) urged down- 
wards by a force equal to the weight of a 
column of the liquid whose base is equal to the orifice, and 
whose height is h ; denoting this pressure, by jt?, and the 
weight of a unit of volume of the liquid, by 10, we shall 
have, 

p = wah. 

If the element is pressed downwards by its own weight 
alone, this pressure being denoted by p\ we have, 

p' = wah' . 

Dividing the former equation by the latter, member by 
member, we have, 

P _ ^ . 

p' ~ h'' 

that is, the pressures are to each other as the heights h 
and h'. 

Were the element to fall through the small height A', 
under the action of the pressure^/, or its own weight, the 
velocity generated would (Art. 115) be given by the 
equation, 



v' - \/2gh'. 

Denoting the velocity actually generated whilst the ele- 
ment is falling throught the height h\ by v, and recol- 
lecting that the velocities generated in falling through a 



MECHANICS OF LIQUIDS. 



267 



given height, are to each other as the square roots of the 
pressures, we shall have, 



v : v' : : -y/p : -yfft, 



, = •</*.. 



Substituting for v' its value, just deduced, and for — f its 



value, j- t , we have 



P 



(150.) 



Hence, we conclude that a liquid will issue from a very 
small orifice at the bottom of the containing vessel, with a 
velocity equal to that acquired by a heavy body in falling 
freely through a height equal to the depth of the orifice 
below the surface of the fluid. 

We have seen that the pressure due to the weight of a 
fluid upon any point of the surface of the containing vessel, 
is normal to the surface, and is always proportional to the 
depth of the point below the level of the free surface, 
Hence, if the side of a vessel be thin, so as not to affect the 
flow of the liquid, and an orifice be made at any point, the 
liquid will flow out in a jet, normal to the surface at the 
opening, and with a velocity due to a height equal to that 
of the orifice from the free surface of the fluid. 

If the orifice is on the vertical side of a vessel, the initial 
direction of the jet will be horizontal ; if it be made at a 
point where the tangent plane is oblique to the horizon, the 
initial direction of the jet will be oblique ; if the opening is 
made on the upper side of a por- 
tion of a vessel where the tangent 
is horizontal, the jet will be 
directed upwards, and will rise 
to a height due to the velocity ; 
that is, to the height of the 
upper surface of the fluid. This 






iy 



\B 



Fig. 153. 



268 MECHANICS. 

can be illustrated experimentally, by introducing a tube near 
the bottom of a vessel of water, and bending its outer 
extremity upwards, when the fluid will be observed to rise 
to the level of the upper surface of the water in the vessel. 

Spouting of Liquids on a Horizontal Plane. 

174. Let KL represent a vessel filled with water. Let 
D represent an orifice in its ver- 
tical side, and DE the path ^ 
described by the spouting fluid. FzB-T^hj 
We may regard each drop of mm ^V\ 
water as it issues from the orifice, 113 p"-y---;iS\ \ 
as a projectile shot forth hori- pHr^-'-^S^A '• 

zontally, and then acted upon by ' ^ ~4r~E' 

the force of gravity. Its path F . 

will, therefore, be a parabola, 

and the circumstances of its motion will be made known by 

a discussion of Equations (115) and (120). 

Denote the distance DK, by A', and the distance Z>X, by 
h. We have, from Equation (120), by making y equal to 
h\ and x = KE, 



KE 

9 



v-ir 



But we have found that v = -\T2gh ; hence, by substitu- 
tion, we have, 

KE = 2y M 7 . 

If we describe a semicircle on KL, as a diameter, and 
through D draw an ordinate DH, we shall have, from a 
well-known property of the circle, 

DH = y/DK.DZ = -y/M 7 . 
Hence we have, by substitution, 

KE = 2&W. 



MECHANICS OF LIQUIDS. 269 

Since there are two points on KL at which the ordinate?. 
are equal, it follows that there are two orhices through 
which the fluid will spout to the same distance on the 
horizontal plane; one of these will be as far above the 
centre 0, as the other is below it. 

If the orifice be at (9, midway between K and X, the 
ordinate OS will be the greatest possible, and the range 
KE' will be a maximum. The range in this case will be 
equal to the diameter of the circle LHK, or to the 
distance from the level of the water in the vessel to the 
horizontal plane. 

If a semi-parabola XXTbe described, having its axis ver- 
tical, its vertex at X, and focus at XT, then may every point 
P, within the curve, be reached by two separate jets issuing 
from the side of the vessel; every point on the curve can be 
reached by one, and only one ; whilst points lying without 
the curve cannot be reached by any jet whatever. 

If the jet is directed obliquely upwards by a short pipe 
A (Fig. 153), the path described by each particle will still be 
the arc of a parabola ABC. Since each particle of the 
liquid may be regarded as a body projected obliquely up- 
ward, the nature of the path and the circumstances of the 
motion will be given by Equation ( 115 ). 

In like manner, # discussion of the same equation will 
make known the nature of the path and the circumstances 
of motion, when the jet is directed obliquely downwards by 
means of a short tube. 

Modifications due to extraneous pressure. 

175. If we suppose the upper surface of the liquid, in 
any of the preceding cases, to be pressed by any force, as 
when it is urged downwards by a piston, we may denote the 
height of a column of fluid whose weight is equal to the ex* 
traneous pressure, by h! . The velocity of efflux will then be 
given by the equation, 



v = y/2g(h + A')- 



^70 MECHANICS. 

The pressure of the atmosphere acts equally on the uppei 
surface and the surface of the opening ; hence, in ordinary 
cases, it may be neglected ; but were the water to flow into 
a vacuum, or into rarefied air, the pressure must be taken 
into account, and this may be done by means of the formula 
just given. 

Should the flow take place into condensed air, or into any 
medium which opposes a greater resistance than the atmos- 
pheric pressure, the extraneous pressure would act upwards, 
h' would be negative, and the preceding formula would 
become, 

v = y/2g{h — h% 

Coefficients of Efflux and Velocity. 

176. When a vessel empties itself through a small orifice 
at its bottom, it is observed that the particles of fluid near 
the top descend in vertical lines ; when they approach the 
bottom they incline towards the orifice, the converging lines 
of fluid particles tending to cross each other as they emerge 
from the vessel. The result is, that the stream grows nar- 
rower, after leaving the vessel, until it reaches a point at a 
distance from the vessel equal to about the radius of the 
orifice, when the contraction becomes a minimum, and below 
that point the vein again spreads out. This phenomenon is 
called the contraction of the vein. The cross section at the 
most contracted part of the vein, is not far from T 6 7 4 ¥ of the 
area of the orifice, when the vessel is very thin. If we de- 
note the area of the orifice, by <2, and the area of the least 
cross section of the vein, by a\ we shall have, 

a' =a ka, 

in which Jc is a number to be determined by experiment. 
This number is called the coefficient of contraction. 

To find the quantity of water discharged through an ori- 
fice at the bottom of the containing vessel, in a second, we 
have only to multiply the area of the smallest cross section 



MECHANICS OF LIQUIDS. 271 

of the vein, by the velocity. Denoting the quantity dis- 
charged in one second, by Q\ we shall have, 



This formula is only true on the supposition that the 
actual velocity is equal to the theoretical velocity, which is 
not the case, as has been shown by experiment. The theo- 
retical velocity has been shown to be equal to ^2gh, and 
if we denote the actual velocity, by v\ we shall have, 



in which I is to be determined by experiment ; this value of 
I is slightly less than 1, and is called the coefficie?it of veloc- 
ity. In order to get the actual discharge, we must replace 
■y/^gh by lyjlgh, in the preceding equation. Doing so, 
and denoting the actual discharge per second, by Q, Ave have. 



The product kl, is called the coefficient of efflux. It has 
been shown by experiment, that this coefficient for orifices 
in thin plates, is not quite constant. It decreases slightly, as 
the area of the orifice and the velocity are increased ; and 
it is further found to be greater for circular orifices than for 
those of any other shape. 

If we denote the coefficient of efflux, by m, we have, 



In this equation, h is called the head of water. Hence, 
we may define the head of water to be the distance from 
the orifice to the plane of the upper surface of the fluid. 

The mean value of m corresponding to orifices of from 
\ to G inches in diameter, Avith from 4 to 20 feet head of 



272 MECHANICS. 

water, has been found to be about .615. If we take the 
value of k = .64, we shall have, 

in .615 

l = k =^ = - 96 - 

That is, the actual velocity is only y 9 ^- of the theoretical 
velocity. This diminution is due to friction, viscosity, &e. 

Efflux through Short Tubes. 

177, It is found that the discharge from a given orifice 
is increased, when the thickness of the plate through which 
the flow takes place, is increased ; also, when a short tube v, 
introduced. 

When a tube AB, is employed which is not more tha^ 
four times as long as the diameter of the 
orifice, the value of m becomes, on an aver- ■ ~>;\//>. I 
age, equal to .813; that is, the discharge ^IjlilP 

per second is 1.325 times greater when the b||Id 

tube is used, than without it. In using the ^ v ^ 5 

cylindrical tube, the contraction takes place 
at the outlet of the vessel, and not at the outlet of the tube. 

Compound mouth-pieces are sometimes used formed of 
two conic frustrums, as shown in the figure, 
having the form of the vein. It has been I 
shown by Etelwein, that the most effec- a\\;;:.'/b~ 

tive tubes of this form should have the /eA 

diameter of the cross section CD, equal ^//^v E 

to .833 of the diameter AB. The angle Fig. ise. 

made by the sides CJEJ and DF, should be 
about 5° 9', and the length of this portion should be three 
times that of the other. 

EXAMPLES. 

1. With what theoretical velocity will water issue from a 
small orifice 16-j 1 ^ feet below the surface of the fluid ? 



MECHANICS OF LIQUIDS. 273 

2. If the area of the orifice, in the last example, is ^ of a 
square foot, and the coefiicient of efflux .615, how many 
cubic feet of water will be discharged per minute ? 

Ans. 118.695 ft. 

3. A vessel, constantly filled with water, is 4 feet high, 
with a cross-section of one square foot ; an orifice in the 
bottom has an area of one square inch. In what time will 
three-fourths of the water be drawn off, the coefficient of 
efflux being .6 ? Ans. -J minute, nearly. 

4. A vessel is kept constantly full of water. How many 
cubic feet of water will be discharged per minute from an 
orifice 9 feet below the upper surface, having an area of 1 
square inch, the coefficient of efflux being .6 ? 

Ans. 6 cubic feet, about. 

5. In the last example, what will be the discharge per 
minute, if we suppose each square foot of the upper surface 
to be pressed by a force of 645 lbs. ? 

Ans. 8-f- cubic feet, about. 

6. The head of water is 16 feet, and the orifice is T £ 7 of 
a square foot. What quantity of water will be discharged 
per second, when the orifice is through a thin plate ? 

SOLUTION. 

In this case, we have, 



Q = .615 x .01/2 x 321 x 16 = .197 cubic feet. 

When a short cylindrical tube is used, Ave have, 

Q = .197 X 1.325 = .261 cubic feet. 

In Etelwein's compound mouthpiece, if we take the 
smallest cross-section as the orifice, and denote it by «, it is 
found that the discharge is 2^ times that through an orifice 
of the same size in a thin plate. In this case, we have, sup 
posing a — T J ¥ of a square foot, 

Q = .197 x 2J = .49 cubic feet. 
12* 



274 MECHANICS. 

Motion of water in open channels. 

178. When water flows through an open channel, as in 
&, river, canal, or open aqueduct, the form of the channel 
being always the same, and the supply of water being con- 
stant, it is a matter of observation that the flow becomes 
uniform ; that is, the quantity of water that flows through 
any cross-section, in a given time, is constant. On account 
of adhesion, friction, &c, the particles of water next the 
sides and bottom of the channel have their motion retarded. 
This retardation is imparted to the next layer of particles, 
but in a less degree, and so on, till a line of particles is 
reached whose velocity is greater than that of any other 
filament. This line, or filament of particles, is called the 
axis of the stream. In the case of cylindrical pipes, the 
axis coincides sensibly with the axis of the pipe; in straight, 
open channels, it coincides with that line of the upper sur- 
face which is midway between the sides. 

A section at right-angles to the axis is called a cross-sec- 
tion, and, from what has been shown, the velocities of the 
fluid particles will be different at different points of the 
same cross-section. The mean velocity corresponding to 
any cross-section, is the average velocity of the particles at 
every point of that section. The mean velocity may be 
found by dividing the volume which flows through the sec- 
tion in one second, by the area of the cross-section. Since 
the same volume flows through each cross-section per 
second, after the flow has become uniform, it follows that, 
in channels of varying width, the mean velocity, at any 
section, will be inversely as the area of the section. 

The intersection of the plane of cross-section with the 
sides and bottom of the channel, is called the perimeter of 
the section. In the case of a pipe which is constantly filled, 
the perimeter is the entire line of intersection of the plane 
of cross-section, with the interior surface of the pipe. 

The mean velocity of water in an open channel depends, 
in the first place, upon its inclination to the horizon. As the 
inclination becomes greater, the component of gravity in the 



MECHANICS OF LIQUIDS. 275 

direction of the channel increases, and, consequently, the 
velocity becomes greater. Denoting the inclination by JT, and 
resolving the force of gravity into two components, one at 
right angles to the upper surface, and the other parallel to 
it, we shall have for the latter component, 

^sinZ 

Tins is the only force that acts to increase the velocity. 
The velocity will be diminished by friction, adhesion, &c. 
The total effect of these resistances will depend upon the 
ratio of the perimeter to the area of the cross section, and 
also upon the velocity. The cross-section being the same, 
the resistances will increase as the perimeter increases ; con- 
sequently, for the same cross-section, the resistance of fric- 
tion will be the least possible when the perimeter is least 
possible. The retardation of the flow will also diminish aa 
the area of the cross-section is increased, other things re- 
maining unchanged. 

If we denote the area of the cross-section by a, the 
perimeter, by P, and the velocity, by v, we shall have, 

^ = m 

m which f denotes some function of v. 

Since the inclination is very small in all practical cases, 
we may place the inclination itself for the sine of the inclin- 
ation, and doing so, it has been shown by Proxy, that the 
function of v may be expressed by two terms, one of which 
is of the first, and the other of the second degree, with re- 
spect to v ; or, 

gal , o 



Denoting — by i?, by k, and - by I, we have, finally, 

if if 

kv + lv 2 = EL 



276 MECHANICS. 

in w Inch k and I are constants, to be determined by experi 
ment. According to Etelwein, we have, 

h = .0000242651, and I = .0001114155. 

Substituting these values, and solving with respect to w, 
we have, 



v ■ = — 0.1088941604 + y.0118580490 -f 8975.414285^7; 

from which the velocity can be found when JR and I are 
known. The values of k and I, and consequently that of v, 
were found by Peony to be somewhat different from those 
given above. Those of Etelwein are selected for the reason 
that they were based upon a much larger number of exper- 
iments than those of Peony. 

Having the mean velocity and the area of the cross- sec- 
tion, the quantity of water delivered in any time can be 
computed. Denoting the quantity delivered in n seconds, 
by §, and retaining the preceding notation, we have, 

Q = nav. 

The quantity of water to be delivered is generally one of 
the data in all practical problems involving the distribution 
of water. The difference of level of the point of supply 
and delivery is also known. The preceding principles ena- 
ble us to give such a form to the cross-section of the canal, 
or aqueduct, as will ensure the requisite supply. 

Were it required to apply the results just deduced, to the 
r ftse of irregular channels, or to those in which there were 
many curves, a considerable modification would be required. 
The theory of these modifications does not come within the 
limits assigned to this treatise. For a complete discussion 
of the whole subject of hydraulics in a popular form, the 
reader is referred to the Traite d'llydr antique D'Atjbisson, 



MECHANICS OF LIQUIDS. 277 

Motion of water in pipes. 

179. The circumstances of the motion of water in pipes, 
are closely analagous to those of its 
motion in open channels. The r- — . 

forces which tend to impart motion |jji 

are dependent upon the weight of ^ Bg§ ^^"""c1jj|ji c 
the water in the pipe, and upon the . ^^ JjllB 

height of the water in the upper Fig . 15T> 

reservoir. Those which tend to 

prevent motion depend upon the depth of water in the 
lower reservoir, friction in the pipe, adhesion, and shocks 
arising from irregularities in the bore of the pipe. The re- 
tardation due to shocks will, for the present, be neglected. 

Let AB represent a straight cylindrical pipe, connecting 
two reservoirs R and H'. Suppose the water to maintain 
its level at E, in the upper, and at (7, in the lower reservoir 
Denote AE, by A, and J3C, by h'. Denote the length of 
the pipe, by I, its circumference, by c, its cross-section, by 
«, its inclination, by <p, and the weight of a unit of volume 
of water, by w. 

Experience shows that, under the circumstances above 
indicated, the flow soon becomes uniform. We may then 
regard the entire mass of fluid in the pipe as a coherent 
solid, moving with a mean uniform velocity down the 
inclined plane AB. 

The weight of the water in the pipe will be equal to ical. 
If we resolve this weight into two components, one perpen- 
dicular to, and the other coinciding with the axis of the 
tube, we shall have for the latter component, loalsimp. But 
/sin? is equal to J)B. Denoting this distance by A", we 
shall have for the pressure in the direction of the axis, due 
to the weight of the water in the pipe, the expression wah". 
This pressure acts from A towards B. The pressure due to 
the weight of the water in J?, and acting in the same 
direction, is wall. 

The forces acting from B towards A, are, first, that dua 



278 MECHANICS. 

to the weight of the water in i?', which is equal to wall' \ 
and, secondly, the resistance due to friction and adhesion. 
This resistance depends upon the length of the pipe, its 
circumference and the velocity. It has been shown, by 
experiment, that this force may be expressed by the term, 

cl(kv -f- &V). 

Since the velocity has been supposed uniform, the forces 
acting in the direction of the axis, must be in equilibrium. 
Hence, 

wah -j- wah" = wall' + cl{kv -f k'v*) ; 

whence, by reduction, 



h h' t ar(h + h"-h'> 

IC w 



V "I V - 1 r- 

c \ I 



The factor - is equal to one-fourth of the diameter of the 

pipe. Denoting this by c?, we shall have, — = \d ; denot- 

h . k' , h+h"-h' c _ 
m g ~ oy m, - by n, and by s, we have, 

mv 4- ?w 2 = -j<&. 

The values of m and n, as determined experimentally by 
Pkony, are, 

m = 0.00017, and n = 0.000106. 

Hence, by substitution, 

.00017^ -f .000106v a =± }ds. 

If v is not very small, the first term may be neglected 
which will give, 

V = 48.56 y/cls. 



MECHANICS OF LIQUIDS. 279 

If we denote the quantity of water delivered in n sec- 
onds, by Q, we shall have, 

Q = nav — 4:8.56na\/ds. 

The velocity will be greatly diminished, if the tube is 
curved to any considerable extent, or if its diameter is not 
uniform throughout. It is not intended to enter into a 
discussion of these cases ; their complete development 
would require more space than has been allotted to this 
branch of Mechanics. 

General Remarks on the distribution and flow of water in pipes. 

180. Whenever an obstacle occurs in the course of an 
open channel or pipe, a change of velocity must take place. 
In passing the obstacle, the velocity of the water will increase, 
and then, impinging upon that which has already passed, a 
shock will take place. This shock consumes a certain 
amount of living force, and thus diminishes the velocity of 
the stream. All obstacles should be avoided ; or, if any are 
unavoidable, the stream should be diminished, and again 
enlarged gradually, so as to avoid, as much as possible, the 
necessary shock incident to sudden changes of velocity. 

For a like reason, when a branch enters the main channel, 
it should be made to enter as nearly in the direction of the 
current as possible. 

All changes of direction give rise to mutual impacts 
amongst the particles, and the more, as the change is more 
abrupt. Hence, when a change of direction is neeessary, 
the straight branches should be made tangential to the 
curved portion. 

The entrance to, and outlet from a pipe or channel, should 
be enlarged, in order to diminish, as much as possible, the 
coefficients of ingress and egress. 

When a pipe passes over uneven ground, sometimes as- 
cending, and sometimes descending, there is a tendency to 
a collection of bubbles of air, at the highest points, which 



280 MECHANICS. 

may finally come to act as an impeding cause to the flow. 

There should, therefore, be suitable pipes inserted at the 

highest points, to permit the confined air to escape. 

Finally, attention should be given to the form of the cross- 

section of the channel. If the channel is a pipe, it should be 

made cylindrical. If it is a canal or open aqueduct, that 

form should be given to the perimeter which would give 

the greatest cross-section, and, at the same time, conform to 

the necessary conditions of the structure. The perimeter in 

open channels is generally trapezoidal, from the necessity 

of the case ; and it should be remembered, that the nearer 

the form approaches a semi-circle, the greater will be the 

flow. 

Capillary Phenomena. 

1§1. When a liquid is in equilibrium, under the action 
of its own weight, it has been shown that its upper surface 
is level. It is observed, however, in the neighborhood of 
solid bodies, such as the walls of a containing vessel, that 
the surface is sometimes elevated, and sometimes depressed, 
according to the nature of the liquid and solid in contact. 
These elevations and depressions result from the action of 
molecular forces, exerted between the particles of the liquid 
and solid which are in contact ; from the fact that they are 
more apparent in the case of small tubes, of the diameter of 
a hair, these phenomena have been called capillary phenom- 
ena, and the forces giving rise to them, capillary forces. 

These forces only produce sensible effects at extremely 
small distances. Clairaut has shown, that when the inten- 
sity of the force of attraction of the particles of the solid for 
those of the liquid, exceeds one-half that of the particles of 
the liquid for each other, the liquid will be elevated about 
the solid ; when less, it will be depressed ; when equal, it 
will neither be elevated nor depressed. In the first case, the 
resultant of the capillary forces is a force of capillary attrac- 
tion ; in the second case, it is a force of capillary repulsion y 
and in the third case, the capillary forces are in equilibrium. 

The following are some of the observed effects of capillary 




MECHANICS OF LIQUIDS. 281 

action : When a solid is plunged into a liquid which ia 
capable of moistening it, as when wood or glass is plunged 
into water, the surface of the liquid is heaped up about the 
solid, taking a concave form, as shown in Fig. 158. 

When a solid is plunged into a liquid which 
is not capable of moistening it, as when 
glass is plunged into mercury, the surface 
of the liquid is depressed about the solid, 
taking a convex form, as shown hi Fig. 159. Fig. 158. 

The surface of the liquid in the neighbor- 
hood of the bounding surfaces of the con- 
taining vessel takes the form of concavity _ 
or convexity, according as the material of 
the vessel is capable of being moistened, , Fi 159 
or not, by the liquid. 

These phenomena become more apparent when, instead of 
a solid body, we plunge a tube into a liquid, according as the 
material of the tube is, or is not, capable of being moistened by 
the liquid, the liquid will rise in the tube or be depressed in 
it. When the liquid rises in the • tube, its upper surface 
takes a concave shape ; when it is depressed, it takes a con- 
vex form. The elevations or depressions increase as the dia- 
meter of the tube diminishes. 



Elevation and Depression between plates. 

182. If two plates of any substance are placed parallel 
to each other, it is found that the laws of ascent and descent 
of the liquid into which they are plunged, are essentially the 
same as for tubes. For example: if two plates of glass 
parallel to each other, and pretty close together, are plunged 
into water, it is found that the water will rise between them 
to a height which is inversely proportional to their dist- 
ance apart ; and further, that this height is equal to half the 
height to which water would rise in a glass tube whose 
internal diameter is equal to the distance between th6 
plates. 





282 MECHANICS. 

If the same plates are plunged into mercury, there vill be 
a depression according to an analagous law. 

If two plates of glass, AB and A C, inclined to each other, 
as shown in Fig. 160, their line of 
junction being vertical, be plunged 
into any liquid which will moisten 
them, the liquid will rise between 
them. It will rise higher near the 

„ , . -, Fig. 160. 

junction, the surface taking a curved 

form, such that any section made by a plane through A, 
will be an equilateral hyperbola. This form of the elevated 
fluid conforms to the laws above explained. 

If the line of junction of the two plates is 
horizontal, a small quantity of a liquid between 
them, which will moisten them, will assume 
the shape shown at A. If the liquid does Fig. i6i. 

not moisten the plates, it will take the form 
shown at B. 

Attraction and Repulsion of Floating Bodies. 

1§3. If two small Balls of wood, both of which can be 
moistened by water, or two small balls of wax, which cannot 
be moistened by water, be placed in a vessel of water, and 
brought so near each other that the surfaces of capillary 
elevation or depression interfere, the balls will attract each 
other and come together. If one ball of wood and one of 
wax be brought so near that the surfaces of capillary eleva- 
tion and depression interfere, the bodies will repel each 
other and separate. If two needles be carefully oiled and 
laid upon the surface of a vessel of water, they will repel 
the water from their neighborhood, and float. If, whilst 
floating, they are brought sufficiently near to each other to 
permit the surfaces of capillary depression to interfere, the 
needles will immediately rush together. The reason of the 
needles floating is, that they repel the water, heaping it up 
on each side, thus forming a cavity in the surface ; the 
needle is buoyed up by a force equal to the weight of the 
displaced fluid, and, when this exceeds the weight of the 



MECHANICS OF LIQUIDS. 283 

needle, it will float. It is on this principle that certain 
insects move freely over the surface of a sheet of water; 
their feet are lubricated with an oily substance which repels 
the water from around them, producing a hollow around 
each foot, and giving rise to a buoyant effort greater than 
the weight of the insect. 

The principle of mutual attraction between bodies, both 
of which repel water, or both of which attract it, accounts 
for the fact that small floating bodies have a tendency to 
collect in groups about the borders of the containing vessel. 
When the material of which the vessel is made, exercises a 
different capillary action from that of the floating particles, 
they will aggregate themselves at a distance from the sur- 
face of the vessel. 

Applications of the Principles of Capillarity. 

184. It is in consequence of capillary action that wator 
rises to fill the pores of a sponge, or of a lump of sugar. 
The same principle, causes the oil to rise in the wick of a 
lamp, which is but a bundle of fibres very nearly in contact, 
leaving capillary interstices between them. 

The siphon filter differs but little in principle from the 
wick of a lamp. It consists of a bundle of fibres like a 
lamp- wick, one end of which dips into a vessel of the liquid 
to be filtered, whilst the other hangs over the edge of the 
vessel. The liquid ascends the fibrous mass by the principle 
of capillary attraction, and continues to advance till it 
reaches the overhanging end, when, if this is lower than the 
upper surface ot the liquid, the liquid will fall by drops from 
the end of the wick, the impurities being left behind. 

The principle of capillary attraction is used for splitting 
rocks and raising weights. To employ this principle in 
cleaving mill-stones, as is done in France, the stone is first 
dressed to the form of a cylinder of the required diameter 
for the mill-stone. Grooves are then cut around it where 
the divisions are to take place, and into these grooves 
thoroughly dried wedges of willow-wood are driven. On 
being exposed to the actior of moisture, the cells of the 



284 MECHANICS. 

wood absorb a large quantity of water, expand, and finally 
split the rock. 

To raise a weight, let a thoroughly dry cord be fastened 
to the weight, and then stretched to a point above.- If, now, 
the cord be moistened, the fibres will absorb the moisture, 
expanding laterally, the rope will be diminished in length, 
and the weight raised. 

The principle of capillary attraction is also very exten- 
sively employed in metallurgy, in a process of purifying 
metals, called cupellation. 

Endosmose and Exosmose. 

1§5. The names endosmose and exosmose have been 
given to two currents flowing in a contrary direction 
between two liquids, when they are separated by a thin 
porous partition, either organic or inorganic. The discovery 
of this phenomena is due to M. Dutrochet, who called the 
flowing in, endosmose, and the flowing out, exosmose. The 
existence of the currents was established by means of an 
instrument, to which he gave the name endosmometre. This 
instrument consists of a long tube of glass, at one end of 
which is attached a membranous sack, secured by a tight 
ligature. If the sack is filled with gum water, a solution of 
sugar, albumen, or, in fact, with almost any solution denser 
than water, and then plunged into water, it is observed, 
after a time, that the fluid rises in the stem, and is depressed 
in the vessel, showing that water has entered the sack by 
passing through the pores. By applying suitable tests, it is 
also found, that a portion c f the liquid in the sack has passed 
through the pores into the vessel. 

Two currents are thus established. If the operation 
be reversed, and the bladder and tube be filled with pure 
water, the liquid in the vessel will rise, whilst that in the 
tube falls. The phenomena of endosmose and exosmose 
are extremely various, and serve to explain a great variety 
of interesting facts in animal and vegetable physiology. 
The cause of the currents is the action of molecular forces 
exerted between the particles of the bodies employed. 



MECHANICS OF GASES AND VAPORS. 285 



CHAPTER VIII. 

MECHANIC? OF GASES AND VAPORS. 

Gases and Vapors. 

186. Gases and vapors are distinguished from other 
fluids, by their great compressibility, and correspondingly 
great elasticity. These fluids continually tend to occupy a 
greater space ; this expansion goes on till counteracted by 
some extraneous force, as that of gravity, or the resistance 
offered by a containing vessel. 

The force of expansion, which is common to all gases and 
vapors, is called their tension or elastic force. We shall 
take for the unit of this force at any point, the pressure 
which would be exerted upon a square inch of surface, were 
the pressure the same at every point of the square inch as 
at the point in question. If we denote this unit, by p, the 
area pressed, by a, and the entire pressure, by J 3 , we shall 
have, 

P = ap (151.) 

Most of the principles already demonstrated for liquids 
hold good for gases and vapors, but there are certain pro- 
perties arising from elasticity which are peculiar to aeriform 
fluids, some of which it is now proposed to investigate. 

Atmospheric Air. 

18?. The gaseous fluid which envelops our globe, and 
extends on all sides to a distance of many miles, is called the 
atmosphere. It consists principally of nitrogen and oxygen, 
together with variable, but small portions of watery vapor 
and carbonic acid, all in a state of mixture. On an average, 
it is found by experiment that 1000 parts by volume of 



286 MECHANICS. 

atmospheric air, taken near the surface of the earth, coi iSists 
of about, 

788 parts of nitrogen, 
197 parts of oxygen, 
14 parts of watery vapor, 
1 part of carbonic acid. 

The atmosphere may, physically speaking, be taken as a 
type of gases, for it is found by experiment that the laws 
regulating the density, expansibility, and elasticity, are the 
same for all gases and vapors, so long as they maintain a 
purely gaseous form. It is found, however, in the case of 
vapors, and of those gases which have been reduced to a 
liquid form, that the law changes just before actual lique- 
faction. 

This change appears to be somewhat analagous to that 
observed when water passes from the liquid to the solid 
form. Although water does not actually freeze till reduced 
to a temperature of 32° Fah., it is found that it reaches its 
maximum density at about 38°. 75, at which temperature the 
particles seem to commence arranging themselves according 
to some new laws, preparatory to taking the solid form. 

Atmospheric Pressure. 
188. If a tube, 35 or 36 inches long, open at one end 
and closed at the other, be filled with pure mercury, and 
inverted in a basin of the same, it is observed 
that the mercury will fall in the tube until the C- 

vertical distance from the surface of the mer- b, 

cury in the tube to that in the basin is about 30 
inches. This column of mercury is sustained by 
the pressure of the atmosphere exerted upon 
the surface of the mercury in the basin, and 
transmitted through the fluid, according to the 
general law of transmission of pressures. The 
column of mercury sustained by the elasticity of Fi 162 
the atmosphere is called the barometric column, 
because it is generally measured by an instrument called a 
barometer. In fact, the instrument just described, when 



& 



MECHANICS OF GASES AND VAPORS. 287 

provided with a suitable scale for measuring the altitude of 
the column, is a complete barometer. The height of the 
barometric column fluctuates somewhat, even at the same 
place, on account of changes of temperature, and other 
causes yet to be considered. 

Observation has shown, that the average height of the 
barometric column at the level of the sea, is a trifle less than 
30 inches. 

The weight of a column of mercury 30 inches in height, 
having a cross section of one square inch, is nearly 15 
pounds. Hence, the unit of atmospheric pressure at the 
level of the sea, is 15 pounds. 

This unit is called an atmosphere, and is often employed 
in estimating the pressure of elastic fluids, particularly in 
the case of steam. Hence, to say that the pressure of steam 
in a boiler is two atmospheres, is equivalent to saying, that 
there is a pressure of 30 pounds upon each square inch of 
the interior of the boiler. In general, when we say that the 
tension of a gas or vapor is n atmospheres, we mean that 
each square inch is pressed by a force of n times 15 pounds. 

Mariotte's Law. 

189. When a given mass of any gas or vapor is com- 
pressed so as to occupy a smaller space, other things being 
equal, its elastic force is increased ; on the contrary, if its 
volume is increased, its elastic force is diminished. 

The law of increase and diminution of elastic force, first 
discovered by Maeiotte, and bearing his name, may be 
enunciated as follows : 

The elastic force of a given mass of any gas, whose tern- 
pen dure remains the same, varies inversely as the volume 
which it occupies. 

As long as the mass remains the same, the density must 
vary inversely as the volume occupied. Hence, from Mari- 
otte's Law, it follows, that, 

The elastic force of any gas, whose temperature remains 
the same, varies as its density, and conversely, the density 
varies as the elastic force. 



Fig. 168 



288 MECHANICS. 

Mariotte's law may be verified in the case of atmospheric 
air, by the aid of an instrument called Maeiotte's Tube. 
This instrument consists of a tube AB CD, of uniform bore, 
bent so that its two branches are parallel to each 
other. The shorter branch AB, is closed at its 
upper extremity, whilst the longer one remains 
open for the reception of mercury. Between the 
two branches of the tube, and attached to the 
same frame with it, is a scale of equal parts for 
measuring distances. 

To use the instrument, place it in a vertical 
position, and pour mercury into the tube, until it 
just cuts off the communication between the two 
branches. The mercury will then stand at the 
same level B C, in both branches, and the tension 
of the confined air in AB, will be exactly equal to that of 
the external atmosphere. If an additional quantity of mer- 
cury be poured into the longer branch, the confined air in 
the shorter branch will be compressed, and the mercury 
will rise in both branches, but higher in the longer, than in 
the shorter one. Suppose the mercury to have risen in the 
shorter branch, to K, and in the longer one, to P. There 
will be an equilibrium in the mercury lying below the hori- 
zontal plane KK\ there will also be an. equilibrium between 
the tension of the air in AK, and the forces which give rise 
to that tension. These forces are the pressure of the exter- 
nal atmosphere transmitted through the mercury, and the 
weight of a column of mercury whose base is the cross-sec- 
tion of the tube, and whose altitude is PK. If we denote 
the height of the column of mercury which will be sustained 
by the pressure of the external atmosphere, by h, the ten- 
sion of the air in AK, will be measured by the weight of a 
column of mercury, whose base is the cross-section of the 
tube, and whose height is A + PK. Since the weight is 
proportional to the height, the tension of the confined air 
will be proportional to h -f PK. 

Now, whatever may be the value of PK, it is found that, 



MECHANICS OF GASES AND VAPORS. 289 

AB . A 



AK = 



A + PK 



B 



If PK = A, we shall have, AK = \AB ; if PK = 2A, 
we shall have, AK = ^4-# ; in general, if P^ = nA, n 
being any positive number, either entire or fractional, we 

AB 

shall have, AK = Maeiotte's Law was verified 

in this manner by Dulong and Akago for all values of n, up 
to n =. 27. The law may also be verified when the pres- 
sure is less than an atmosphere, by means of the following- 
apparatus. 

AK represents a straight tube of uniform bore, closed at 
its upper and open at its lower extremity : CD 
is a long cistern of mercury. The tube AK is n A 

either graduated into equal parts, commencing 
at A, or it has attached to it a scale of brass or 
ivory. 

To use the instrument, pour mercury into the c 

tube till it is nearly full ; place the finger over 
the open end, and invert it in the cistern of mer- 
cury, and depress it till the mercury stands at 
the same level without, as within the tube, and 
suppose the surface of the mercury in this case -pig. 164. 

to cut the tube at B. Then will the tension 
of the confined ah- in AB, be equal to that of the external 
atmosphere. If now the tube be raised vertically, the air in 
AB will expand, its tension will diminish and the mercury 
will fall in the tube, to maintain the equlibrium. Suppose 
the level of the mercury in the tube to have reached 
the point K In this position of the instrument the tension 
of the air in AK, added to the weight of the column of mer- 
cury, KE will be equal to the tension of the external air. 

Now, it is found, whatever may be the value of KE, that 

AB.h 



AK= A 

A — EK 



13 



290 MECHANICS. 

If EK = JA, we have, AK = 2 AS; if EK -^ |A, we 

have, ^liT = 3AB; in general, if EK — A, we have, 

A 7? "w + 1 ' 

n-\-\ 

Maeiotte-s law has been verified in this manner, for all 
values of n, up to n = 111. 

It is a law of Physics that, when a gas is suddenly com- 
pressed, heat is evolved, and when a gas is suddenly ex- 
panded, heat is absorbed ; hence, in making the experiment, 
care must be taken to have the temperature kept uniform. 

Gay Lussac's Law. 

190. If, whilst the volume of any gas or vapor remains 
the same, its temperature be increased, its tension is in- 
creased also. If the pressure remain the same, the volume 
of the gas will increase as the temperature is raised. The 
law of increase and diminution, as deduced by Gay Lussac, 
whose name it bears, may be enunciated as follows : 

In a given mass of any gas, or vapor, if the volume 
remains the same, the tension varies as the temperature ; if 
the tension remains the same, the volume varies as the tem- 
perature. 

According to Regnault, if a given mass of atmospheric 
air be heated from 32° Fahrenheit to 212°, the tension, or 
pressure remaining constant, its volume will be increased by 
the .3665th part of the volume at 32°. Hence, the increase 
of volume for each degree of temperature is the .00204th part 
of the volume at 32°. If we denote the volume at 32° by v, 
and the volume at the temperature t ', by v', we st all there- 
fore have, 

v' = v[l + .00204(£'- 32)] . . ( 152.) 

Solving with reference to v, we have, 

v' 
1 + .00204(£'- 32) v ' '/ 

Formula (153) enables us to compute the volume of any 



MECHANICS OF GASES AND VAPOUS. 



291 



mass of air at 32°, knowing its volume at the temperature 
f, the pressure remaining constant. 

To find the volume at the temperature t'\ we have simply 
to substitute t" for t' in (152.) Denoting this volume by 
v", we have, 

v"= v[l + .002040"- 32)]. 



Substituting for v its value from (153), we get, 



v = v 



,1 4- .00204(£" — 32) 
1 + .00204(£' - 32) 



(154.) 



This formula enables us to compute the volume of any 
mass of air, at a temperature t", when we know its volume 
at the temperature t' ; and, since the density varies in- 
versely as the volume, we may also, by means of the same 
formula, fh\d the density of any mass of air, at the temper- 
ature t'\ when we have given its density at the tempera- 
ture t'. 

Manometers. 

191. A manometer is an instrument used for measuring 
the tension of gases and vapors, and particularly of steam. 
Two principle varieties of manometers are used for measur- 
ing the tension of steam, the open manometer, and the 
closed manometer. 

The open Manometer. 

192. The open manometer consists, essentially, of an 
open glass tube AJB, terminating below, 

nearly at the bottom of a cistern EF. 
The cistern is of wrought iron, steam 
tight, and filled with mercury, Its dimen- 
sions are such, that the upper surface of 
the mercury wall not be materially lowered, 
when a portion of the mercury is forced 
up the tube. ED is a tube, by means of 
which, steam may be admitted from the 
boiler to the surface of the mercury in the 
cistern. This tube is sometimes filled with Fig. 165. 



Arm 




292 MECHANICS. 

water, through which the pressure of the steam is trans 
mitted to the mercury. 

To graduate the instrument. All communication with 
the boiler is cut off, by closing the stop-cock JEJ, and commu- 
nication with the external air is made by opening the stop- 
cock D, The point of the tube AjB, to which the mercury 
rises, is noted, and a distance is laid off, upwards, from this 
point, equal to what the barometric column wants of 30 
inches, and the point .ZTthus determined, is marked 1. This 
point will be very near the surface of the mercury in the 
cistern. From the point H, distances of 30, 60, 90, &c, 
inches are laid off upwards, and the corresponding points 
numbered 2, 3, 4, &c. These divisions correspond to 
atmospheres, and may be subdivided into tenths and 
hundredths. 

To use the instrument, the stop-cock D is closed, and a 
communication made with the boiler, by opening the stop- 
cock E. The height to which the mercury rises in the 
tube, will indicate the tension of the steam in the boiler, 
which may be read from the scale in terms of atmospheres 
and decimals of an atmosphere. If the pressure in pounds 
is wished, it may at once be found, by multiplying the 
reading of the instrument by 15. 

The principal objection to this kind of manometer, is its 
want of portability, and the great length of tube required, 
when high tensions are to be measured. 

The closed Manometer. 

? 9r. The general construction of the closed manometer 
is the same as that of the open manometer, with the excep- 
tion that the tube AB is closed at the top. The ah- which 
is confined in the tube, is then compressed in the same way 
as in Mariotte's tube. 

To graduate this instrument. We determine the division 
H, as before. The remaining divisions are found by apply- 
ing Mariotte's law. 

Denote the distance in inches, from H to the top of the 



MECHANICS OF GASES AND VAPORS. 293 

lube, by I; the pressure on the mercury, expressed in 
atmospheres, by n, and the distance in inches, from H to the 
upper surface of the mercury in the tube, by x. 

The tension of the air in the tube will be equal to that on 
the mercury in the cistern, diminished by the weight of a 
column of mercury, whose altitude is x. Hence, in atmos- 
pheres, it is 

x 

The bore of the tube being uniform, the volume occupied 

by the compressed air will be proportional to its height. 

When the pressure is 1 atmosphere, the height is l\ when 

x 
the pressure is n atmospheres, the height is I — x. 

Hence, from Mariotte's law, 

1: »-£.:: I-a: I. 

Whence, by reduction, 

x 3 — (30rc + l)x = — S0l(n — 1). 
Solving, with respect to sc, we have, 



_ ± y/_ 30J(» - 1) + {-—-) ' 



The upper sign of the radical is not used, as it would give 
a value for x, greater than I. Taking the lower sign, and, as 
a particular case, assuming I = 30 in., we have, 



x = 15n + 15 — </ — 900(^ — 1) + (15n + 15) 2 . 

Making n = 2, 3, 4, <fcc., in succession, we find for aj, the 
corresponding values, 11.46 in., 17.58 in., 20.92 in., &c. 
These distances being set off from H, upwards, and marked 
2, 3, 4, &c, indicate atmospheres. The intermediate spaces 
are subdivided by means of the same formula. 



294 



MECHANICS, 



Fig. 166. 



The use of this instrument is the same as that of the. 
manometer last described. 

In making the graduation, we have supposed the tem- 
perature to remain the same. If, however, it does not 
remain the same, the reading of the instrument must be 
corrected by means of a table computed for the lurpose. 

The instruments already described, can only be used for 
measuring tensions greater than one atmosphere. 

The Siphon Guage. 

194. The siphon guage is an instrument employed to 
measure tensions of gases and vapors, 

when they are less than an atmosphere. 
It consists of a tube AB 6 r , bent so that 
its two branches are parallel. The branch 
B G is closed at the top, and filled with 
mercury, which is retained by the pres- 
sure of the atmosphere, whilst the branch 
AB is open at the top. If, now, the air 
be rarified in any manner, or if the mouth 
A of the tube, be exposed to the action of any gas whose 
tension is sufficiently small, the mercury will no longer be 
supported in the branch B (7, but will fall in that and rise in 
the other. The distance between the surfaces of the mer- 
cury in the two branches, as given by a scale placed between 
them, will indicate the tension of the gas. If this distance 
is expressed in inches, the tension can be found, in atmos- 
pheres, by dividing by 30, or, in pounds, by dividing by 2. 

The Diving-Bell. 

195. The diving-bell is a bell-shaped vessel, open at 
the bottom, used for descending below the 

surface of the water. The bell is placed 
bo that its mouth shall continue horizontal, 
and is let down by means of a rope AB, 
and the whole apparatus is sunk by 
weights properly adjusted. The air con- 
tained in the bell before immersion, will 
be compressed by the weight of the 




Fig 167. 



MECHANICS OF GASES AND VAPORS. 295 

water, but its increased elasticity will prevent the water 
from rising to the top of the bell, which is provided with 
seats for the accommodation of those wishing to descend. 
The air within is constantly contaminated by breathing, and 
is continually replaced by fresh air, pumped in through a 
tube FG. Were there no additional air introduced, the 
volume of the compressed air, at any depth, might be com- 
puted by Mariotte's law. The unit of the compressing 
force, in this case, is the w r eight of a column of water whose 
cross-section is a square inch, and whose height is the 
distance from D 6 Y , to the surface of the water. 
The Barometer. 

196. The barometer is an instrument for measuring 
the pressure of the atmosphere. As already explained, it 
consists of a glass tube, hermetically sealed at one extre- 
mity, which is filled with mercury, and inverted in a basin 
of that fluid. The pressure of the air is indicated by the 
height of the column of mercury which it supports. 

A great variety of forms of the mercurial barometer have 

been devised, all involving the same mechanical principle. 

The two most important of these are the siphon and the 

cistern barometer. 

The Siphon Barometer. 

197. The siphon barometer consists essentially of a 
tube CDE, bent so that its two branches, CD 

and DE, shall be parallel to each other. A 

scale of equal parts is placed between them, 

and attached to the same frame with the tube. 

The longer branch CD, is about 32 or 33 

inches in length, hermetically sealed at the top, 

and filled with mercury ; the shorter one is 

open to the action of the air. When the 

instrument is placed vertically, the mercury 

sinks in the longer branch and rises in the -^ 

shorter one. The distance between the sur- Flg * 16S * 

face of the mercury in the two branches, as measured by 

the scale of equal parts, indicates the pressure of the atmoa 

phere at the particular time and place. 



I 



296 



MECHANICS, 






The Cistern Barometer. 

198. The cistern barometer consists of a glass tube 
Med and inverted in a cistern of mercury, as already 
explained. The tube is surrounded by a frame of metal, 
firmly attached to the cistern. Two opposite longitudinal 
openings, near the upper part of the frame, permit the 
upper surface of the mercury to be seen. A slide, moved 
up and down by means of a rack and pinion, may be 
brought exactly to the upper level of the mercury. The 
height of the column is then read from a scale, so adjusted as 
to have its at the surface of the mercury in the cistern. 
The scale is graduated to inches and tenths, and the smaller 
divisions are read by means of a vernier. 

The figure shows the arrangement of parts in 
a complete cistern barometer. KK represents 
the frame of the barometer ; HH that of the 
cistern, open at the upper part, that the level 
of the mercury in the cistern may be seen 
through the glass; X, an attached thermo- 
meter, to show the temperature of the mer- 
cury in the tube ; N, a part of the sliding ring 
bearing the vernier, and moved up and down 
by the milled-headed scEew M. 

The particular arrangement of the cistern is 
shown on an enlarged scale in Fig. 110. A 
rej)resents the barometer tube, terminating in 
a small opening, to prevent too sudden shocks 
when the instrument is moved from place to 
place ; H represents the frame of the cis- 
tern; B, the upper portion of the cistern, 
made of glass, that the surface of the mercury 
may be seen ; E, a conical piece of ivory, pro- 
jecting from the upper surface of the cistern : 
when the surface of the mercury just touches 
the point of the ivory, it is at the of the 
scale; GO represents the lower part of the 
cistern, and is made of leather, or some other 



HI 



MECHANICS OF GASES AND VAPORS. 297 

flexible substance, and firmly attached to the glass part* 
D is a screw, working through the bottom of the frame, and 
against the bottom of the bag (7(7, through the medium of 
a plate JP. The screw D, serves to bring the surface of the 
mercury to the point of the ivory piece JE^ and also to force 
the mercury up to the top of the tube, when it is desired 
to transport the barometer from place to place. 

To use this barometer, it should be suspended vertically, 
and the level of the mercury in the cistern brought to the 
point of the ivory piece E, by means of the screw D ; \ a 
smart rap with a key upon the frame will detach the mer- 
cury from the glass to which it sometimes tends to adhere. 
The sliding ring iV, is next run up or down by means of the 
screw J/, till its lower edge appears tangent to the upper 
surface of the mercury in the tube, and the altitude is read 
from the scale. The height of the attached thermometer 
should also be noted. 

The requirements of a good barometer are, sufficient 
width of tube, perfect purity of the mercury, and a scale 
with a vernier accurately graduated and adjusted. 

The bore of the tube should be as large as practicable, to 
diminish the effect of capillary action. On account of the 
mutual repulsion between the particles of the glass and mer- 
cury, the mercury is depressed in the tube, and this depres- 
sion increases as the diameter of the tube diminishes. 

In all cases, this depression should be allowed for, and 
corrected by means of a table computed for the purpose. 

To secure purity of the mercury, it should be carefully 
distilled, and after the tube is filled, it should be boiled over 
a spirit-lamp, to drive off any bubbles of air that might ad- 
here to the walls of the tube. 

Uses of the Barometer. 

199. The primary object of the barometer is, to meas- 
ure the pressure of the atmosphere at any time or place. It 
is used by mariners and others, as a weather-glass. It is 
also extensively employed for determining the heights of 
points on the earth's surface, above the level of the ocean. 
]3* 



298 



MECHANICS. 



The principle on which it is employed for the latter pur 
pose is, that the pressure of the atmosphere at any place 
depends upon the weight of a column of air reaching from 
the place to the upper limit of the atmosphere. As we as- 
cend above the level of the ocean, the weight of the column 
diminishes ; consequently, the pressure becomes less, a fact 
which is shown by the mercury falling in the tube. We 
shall investigate a formula for determining the difference of 
level between any two points. 

Difference of Level. 

200. Let aB represent a portion of a vertical prism of 
air, whose cross-section is one square inch. De- 
note the pressure on the lower base JB, by />, and 
on the upper base aa\ by p' ; denote the density «=— t 
of the air at B, by d, and at aa\ by d', and sup- 
pose the temperature throughout the column to 
be 32° Fah. 

Pass a horizontal plane bb\ infinitely near to 
aa\ and denote the weight of the elementary Fl 171 
volume of air ab, by w. Conceive the entire 
column to be divided by horizontal planes into elementary 
prisms, such that the weights of each shall be equal to w, 
and denote their heights, beginning at a, by s, s', s", &c. 

From Mariotte's law, we shall have, 

£-■-■—€-. 

p ~ d 

The air throughout each elementary prism may be re- 
garded as homogeneous ; hence, the density of the air in ab 
is equal to its weight, divided by its volume into gravity 
(Art. 12). But its volume is equal to lxlxs^s 
hence, 

9* 
Substituting this in the preceding equation, we have, 



u 



MECHANICS OF GASES AND VAPORS. 299 



whence, 



P _ w 
p ~~ gsd' 



s = -f- X - • • • • (155.) 
dg p' 



From Davies' Bourdon, page 297, we have, by substitut- 

w 
ing for y the fraction — , the equation, 

/ w\ w w* w % s 

But — being infinitely small, all the terms in the second 
member, after the first, may be neglected, giving, 

or finally, 

« = l(p> + v>) - lp', 

in which I denotes the Napierian logarithm. 

In this equation, p' denotes the pressure on the prism ab ; 

hence, p' + w denotes the pressure on the next prism 

below, that is, on the prism be. 

w 
If sve substitute this value of — in Equation (155), we 

shall have, for the height of the prism ab, 

s = §- g [Kp'+«>)-¥l 

Substituting in succession for p\ the values p' + w, p' -\- 2to, 
p' 4- 3?c, &c., we shall find the heights of the elementary 
prisms be, ed, &v.. We shall therefore have, 



300 MECHANICS. 



»'= g[V+2 W )-*(y + «,)], 



*" = §zV{P'+ 3«>) - l(p' + 2w)l 



If w denote the number of elementary prisms in AJB, the 
sum of the first members will be equal to AJB. Adding the 
equations member to member, and denoting the sum of the 
first members by z, we have, 

Because nw denotes the weight of the column of air A B, 
we shall have, p' + nw — p, hence, 

8=.#:* £ , (156.) 

dg p' 

Denoting the modulus of the common system of loga- 
rithms by M, and designating common logarithms by the 
symbol log, we shall have, 

Mz = ^- log —., or z — ~r 1°£ ^ * 
dg s p" Mdg B p' 

Now, the pressures jp and// are measured by the heights of 
the columns of mercury which they will support ; denoting 
these heights by -£Tand H\ we have, 

E - K 
p' ~ H' ' 



MECHANICS OF GASES AND VAPORS. 301 

whence, by substitution, 

We have supposed the temperature, both of the air and 
mercury, to be 32°. In order to make the preceding for- 
mula general, let T represent the temperature of the mer- 
cury at J5,. T\ its temperature at a r .and. denote the cor- 
responding heights of the barometric column by h and h' ; 
also, let t denote the temperature of the air at B, and t' its 
temperature at a. 

The quantity - is the ratio of the density of the ah* at JB y 

to the corresponding pressure, the temperature being 32°. 
According to Mariotte's law, this ratio remains constant, 
whatever may be the altitude of B ab<ne the level of the 
ocean. 

If we denote the latitude of the ph<iQ by I, we have, 
(Art. 124), 

g - g'(l — 0.002695 cos2 7 \ 

It has been shown, by experiment, that, vt^n a column 
of mercury is heated, it increases in length at thp ra + e of 
&9*9o tDS °f * ts l en gth at 32°, for each degree th?-t the tem- 
perature is elevated. Hence, 



h - fl/i 4- T ~ S2 \ ^9990+ T- 
h = S V + -9990-; = H ^90 



32 



? 



-32 



Dividing the second equation by the first, member hy 
member, 

h_ _ H_ 9990 + T— 32 
h' ~~ ~W 9990 + T— 32 " 



302 MECHANICS. 

TT 

Dividing both terms of the fractional coefficient of -3= bj 

the denominator, and neglecting the quantity T — 32, in 
comparison with 9990, we have, 

Whence, by reduction, 

IT h 1 



H' h' 1 + .0001(r- V) 

The quantity z denotes, not only the height, but also the 
volume of the column of air aB, at 32°. When the tem- 
perature is changed from 32°, the pressures remaining the 
same, this volume will vary, according to the law of Gay 
Lussac. 

If we suppose the temperature of the entire column to be 
a mean between the temperatures at B and a, which we 
may do without sensible error, the height of the column 
will become, Equation (153), 

z |"l + .00204 (^tl _ 32 VI = z[l +.00102(2 + 2'- 64)] 

Hence, to adapt Equation (157) to the conditions pro- 
posed, we must multiply the value of z by the factor, 

1 + .00102(2 + t' — 64). 

TT 

Substituting in Equation (157), for -=-, and g, the values 

shown above, and multiplying the resulting value of 2, by 
the factor 1 + .00102(2 + 2' — 64), we have, 

_ P 1 + .00102(2 + 2' — 64) h 

Z ~ Md* 1 - 0.002695cos2/ ° g h'[l + .OOOIJT-T 1 )] 

(158.) 



MECHANICS OF GASES AND VAPORS. 303 

P 

The factor -^ is constant, and may be determined as 

follows: select two points, one of which is considerably 
higher than the other, and determine, by trigonometrical 
measurement, their difference of level. At the lower point, 
take the reading of the barometer, of its attached ther- 
mometer, and of a detached thermometer exposed to the 
air. Make similar observations at the upper station. These 
observations, together with the latitude of the place, will 
give all the quantities entering Equation (158), except the 
factor in question. Hence, this factor may be deduced. It 
is found to be 60345.51 ft. Hence, we have, finally, the 
barometric formula, 

z — 60345.51 ft. X 

3 +.00102 (* + f- 84) h 

l-0.002695cos2J g h'\\ + .0001(r~ T')] V ' 

To use this formula for determining the difference of level 
between two stations, observe, simultaneously, if possible, 
the heights of the barometer and of the attached and de- 
tached thermometers, at the two stations. Substitute these 
results for the corresponding quantities in the formula ; also 
substitute for I the latitude of the place, and the resulting 
value of 2, will be the difference of level required. 

If the observations cannot be made simultaneously at the 
two stations, make a set of observations at the lower station ; 
after a certain interval, make a set at the upper station ; 
then, after an equal interval, make another set at the lower 
station. Take a mean of the results of observation at the 
lower station, as a single set, and proceed as before. 

For the more convenient application of the formula for 
the difference of level between two points, tables have been 
computed, by means of which the arithmetical operations 
are much facilitated. 



ll 



#04 MECHANICS. 

Work due to the Expansion of a Gas or Vapor. 

201. Let the gas or vapor be confined in a cylindei 

closed at its lower end, and having 
a piston working air-tight. When 
the gas occupies a portion of the 
cylinder whose height is h, denote the 
pressure on each square inch of the 
piston by p ; when the gas expands, 
so that the altitude of the column be- 
comes x, denote the pressure on a jl" 
square inch by y. p ig . 172. 

Since the volumes of the gas, under 
these suppositions, are proportional to their altitudes, we 
shall have, from Mariotte's laws, 

p : y : ; x : h ; 
whence 

xy = ph 

If we suppose p and h to be constant, and x and y to 
vary, the above equation will be that of an equilateral 
hyperbola referred to its asymptotes. 

Draw A C perpendicular to AM, and on these lines, as 
asymptotes, construct the curve NIjH, from the equation, 
xy = ph. Make AG — h, and draw GH parallel to AC; 
it will represent the pressure p. Make AM = x, and draw 
MN parallel to A G ; it will represent the pressure y. In 
like manner, the pressure at any elevation of the piston may 
be constructed. 

Let KL be drawn infinitely near to GH, and parallel 
with it. The elementary area GKLH will not differ 
sensibly from a rectangle whose base is p, and altitude is 
GK. Hence, its area may be taken as the measure of the 
work whilst the piston is rising through the infinitely small 
space GK. In like manner, the area of any infinitely small 
element, bounded by lines parallel to A (7, may be taken tc 
represent the work whilst the piston is rising through the 



MECHANICS OF GASES AND VAPORS. 305 

height of the element. If we take the sum of all the 
elements between the ordinates Gil and ]!£]¥, this sum, or 
the area GMNH, will represent the total quantity of work 
of the force of expansion whilst the piston is rising from G 
to M. But the area included between an equilateral hyper- 
bola and one of its asymptotes, and limited by lines parallel 
to the other asymptote, is equal to the product of the co- 
ordinates of any point, multiplied . by the Naperian 
logarithm of the quotient obtained by dividing one of the 
limiting ordinates by the other ; or, in this particular case, 

it is equal to ph x l( — ]. Hence, if we designate the 

quantity of work performed by the expansive force whilst 
the piston is moving over GM, by q, we shall have, 

This is the quantity of work exerted upon each square inch 
of the piston ; if we denote the area of the piston, by A, 
and the total quantity of work, by Q, we shall have, 

Q = Aph x l(P) = Aph x?(|) . ( 160.) 

If we denote by c the number of cubic feet of gas, when the 

pressure is p, and suppose it to expand till the pressure is y, 

we shall have, Ah — c ; or, if A be expressed in square 

Ah 

feet, we shall have, c = Hence, by substitution, 

'144 » J ' 



Q = U4cpxl(?-) 



y 

Finally, if we suppose the pressure at the highest point to 
be p\ we shall have, 



« = 144<jpxl(J), 



306 MECHANICS. 

an equation which gives the quantity of work ot c cubic 

feet of gas, whilst expanding from a pressure p, to a pres« 

sure p'. 

Efflux of a Gas or Vapor. 

202. Suppose the gas to escape from a small orifice, and 
denote its velocity by v. Denote the weight of a cubic 
foot of the gas, by w, and the number of cubic feet dis- 
charged in one second, by c, then will the mass escaping in 

cw 
one second, be equal to — , and its living force will be 

cw 
equal to — v 2 . But, from Art. 148, the living force is 

if 

double the accumulated quantity of work. If, therefore, we 
denote the accumulated work by Q, we shall have, 

r\ cw 2 
* 2g 

But the accumulated work is due to the expansion of the 
gas, and if we denote the pressure within the orifice, by p, 
and without, by^/, we shall have, from Art. 201, 



q = iucp x i&y 



P' 

Equating the second members, we have, 
Whence, 



v = 12 



v^Rf) 



Substituting for g, its value, 32£ ft., we have, aftei 
reduction, 



^\4 X <?) • • • < w ?-> 



MECHANICS OF GASES AND VAPORS. 307 

When the difference between p and p' is small, the pre- 
ceding formula can be simplified. 

Since — = 1 + ^ ~? , we have, from the logarithmic 
p p 

series, 

When jo — p' is very small, the second, and all succeeding 
terms of the development, may be neglected, in comparison 
with the first term. Hence, 

\p'J p' 

Substituting, in the formula above deduced, we have, 



v = w Jp x p-£. 

V w p 



or, since — r is, under the supposition just made, equal to 1, 
we have, finally, 



V = 96 V /^— ^ (162.) 

V w 



Coefficient of Efflux. 

203. When air issues from an orifice, the section of the 
current undergoes a change of form, analagous to the con- 
traction of the vein in liquids, and for similar reasons. If 
we denote the coefficient of efflux, by &, the area of the 
orifice, by A, and the quantity of air delivered in n seconds, 
by Q, we sna ^ have, from Equation (161), 



e = M*-viKi?) 



308 MECHANICS. 

According to Koch, the value of h is equal to .58, when 
the orifice is in a thin plate ; equal to .74, when the air 
issues through a tube 6 times as long as it is wide ; and 
equal to .85, when it issues through a conical nozzle 5 times 
as long as the diameter of the orifice, and whose sides have 
a convergence of 6° to the axis. 

The preceding principles are applicable to the distribution 
of gas, to the construction of blowers, and, in general, to a 
great variety of pneumatic machines. 

Steam. 

204. If water be exposed to the atmosphere, at ordinary 
temperatures, a portion is converted into vapor, which mixes 
with the atmosphere, constituting one of the permanent 
elements of the aerial ocean. The tension of watery vapor 
thus formed, is very slight, and the atmosphere soon ceases 
to absorb any more. If the temperature of the water be 
raised, an additional amount of vapor is evolved, and of 
greater tension. When the temperature is raised to that 
point at which the tension of the vapor is. equal to that of 
the atmosphere, ebullition commences, and the vaporization 
goes on with great rapidity. If heat be added beyond the 
point of ebullition, neither the water nor the vapor will 
increase in temperature till all of the water is converted into 
steam. "When the barometer stands at 30 inches, the boil- 
ing point of pure water is 212° Fah. We shall suppose, in 
what follows, that the barometer stands at 30 inches. After 
the temperature of the water is raised to 212°, the addi- 
tional heat that is added becomes latent in the vapor 
evolved. 

If heat be applied uniformly, it is found by experiment 
that it takes b\ times as much to convert all of the water 
into steam as it requires to raise it from 32° to 212°. Hence, 
the entire amount of heat which becomes latent is 
5J X (212° — 32°) =; 990°. That the heat applied becomes 
latent, may be shown experimentally as follows : 

Let a cubic inch of water be converted into steam at 



MECHANICS OF GASES AND VATORS. 309 

212°, and kept in a close vessel. Now, if b\ cubic inches 
of water at 32° be injected into the "vessel, the steam will all 
be converted into water, and the 6^ cubic inches of water 
will be found to have a temperature of 212°. The heat 
that was latent becomes sensible again. 

When water is converted into steam under any other 
pressure than that of the atmosphere, or 15 pounds to the 
square inch, it is found that, although the boiling point will 
be changed, the entire, amount of heat required for convert- 
ing the water into steam will remain unchanged. 

If the evaporation takes place under such a pressure, that 
the boiling point is but 150°, the amount of heat which 
becomes latent is 1052°, so that the latent heat of the 
steam, plus its sensible heat, is 1202°. If the pressure under 
which vaporization takes place is such as to raise the boiling 
point to 500°, the amount of heat which becomes latent is 
'702°, the sum 702° + 500° being equal to 1202°, as before. 
Hence, we conclude that the same amount of fuel is 
required to convert a given amount of water into steam, no 
matter what may be the pressure under which the evapora- 
tion takes place. 

When water is converted into steam under a pressure of 
one atmosphere, each cubic inch is expanded into about 
1700 cubic inches of steam, of the temperature of 212° ; or, 
since a cubic foot contains 1728 cubic inches, we may say, 
in round numbers, that a cubic inch of water is converted 
into a cubic foot of steam. 

If water is converted into steam under a greater or less 
pressure than one atmosphere, the density will be increased 
or diminished, and, consequently, the volume will be dimin- 
ished or increased. The temperature being also increased 
or diminished, the increase of density or decrease of volume 
will not be exactly proportional to the increase of pressure ; 
but, for purposes of approximation, we may consider the 
densities as directly, and the volumes as inversely propor- 
tional to the pressures under which the steam is generated. 
Under this hypothesis, if a cubic inch of water be evapo- 



310 



MECHANICS. 



rated under a pressure of a half atmosphere, it will afford 

two cubic feet of steam ; if generated under a pressure of 

two atmospheres, it will only afford a half cubic foot of steam. 

Work of Steam. 

205. When water is converted into steam, a certain 
amount of work is generated, and, from what has been shown, 
this amount of work is very nearly the same, whatever may 
be the temperature at which the water is evaporated. 

Suppose a cylinder, whose cross-section is one square 
inch, to contain a cubic inch of water, above which is an air- 
tight piston, that may be loaded Avith weights at pleasure. 
In the first place, if the piston is pressed down by a weight 
of 15 pounds, and the inch of water converted into steam, 
the weight will be raised to the height of 1728 inches, or 
144 feet. Hence, the quantity of work is 144 x 15, or, 
2160 units. Again, if the piston be loaded with a weight 
of 30 pounds, the conversion of water into steam will give 
but 864 cubic inches, and the weight will be raised through 
72 feet. In this case, the quantity of work will be 72 x 30, 
or 2160 units, as before. We conclude, therefore, that the 
quantity of work is the same, or nearly so, whatever may be 
the pressure under which the steam is generated. We also 
conclude, that the quantity of work is nearly proportional to 
the fuel consumed. 

Besides the quantity of work developed by simply con- 
verting an amount of water into steam, a further quantity 
of work is developed by allowing the steam to expand after 
entering the cylinder. This principle is made use of in 
steam engines working expansively. 

To find the quantity of work developed by steam acting ex- 
pansively. Let A JB represent a cylinder, closed at ' 
A, and having an air-tight piston D. Suppose the 
steam to enter at the bottom of the cylinder, and to B 
push the piston upward to (7, and then suppose 
the opening at which the steam enters, to be j 
closed. If the piston is not too heavily loaded, 



the steam will continue to expand, and the piston Fi m 



MECHANICS OF GASES AND VAPOKS. 311 

will be raised to some position, B. The expansive force 
of the steam will obey Maeiotte's law, and the quantity of 
work due to expansion will be given by Equation (160). 

Denote the area of the piston in square inches, by A ; the 
pressure of the steam on each square inch, up to the moment 
when the communication is cut off, by p ; the distance A C, 
through which the piston moves before the steam is cut oflj 
by h ; and the distance AD, by nh. 

If we denote the pressure on each square inch, when the 

piston arrives at B, by p\ we shall have, by Mariotte's 

law, 

P 
p : p' : : nh : h, . • . p' = — , 

n 

an expression which gives the limiting value of the load of 
the piston. 

The quantity of work due to expansion being denoted by 
£, we shall have, from Equation (160), 

q = Aph x I (-J-) = Aphl (n). 

If we denote the quantity of work of the steam, whilst 
the piston is rising to C, by q'\ we shall have, 

q" = Aph. 

Denoting the total quantity of work during the entire stroke 
of the piston, by Q, we shall have, 

Q = Aph [1 + l(n)] . . . ( 163.) 

Experimental Formulas. 

206. Numerous experiments have been made for the 
purpose of determining the relation existing between the 
elasticity and temperature of steam in contact with the 
water by which it is produced, and many formulas, based 



812 MECHANICS. 

upon these experiments, have been given, two of which art 
subjoined : 

The formula of Dulong and Arago is, 

p = (1 + .007153*) 3 , 

in which p represents the tension in atmospheres, and t the 
excess of the temperature above 100° Centigrade. 
Tredgold's formula is, 

t = 0.85-t/p — 75, 

in which t is the temperature, in degrees of the Centigrade 
thermometer, and p the pressure, expressed in centimeters 
of the mercurial column. 



HYDRAULIC AND PNEUMATIC MACHINES. 



313 



CHAPTER IX. 



HYDRAULIC AND PNEUMATIC MACHINES. 



Definitions. 

207. Hydraulic machines are those used in raising and 
distributing water, such as pumps, siphons, hydraulic rams, 
&c. The name is also applied to those machines in which 
water power is the motor, or in which water is employed to 
transmit pressures, such as water-wheels, hydraulic presses, &c. 

Pneumatic machines are those employed to rarefy and 
condense air, or to impart motion to the air, such as air- 
pumps, ventilating-blowers, &c. The name is also applied 
to those machines in which currents of air furnish the motive 
power, such as windmills, &c. 

Water Pumps. 

20§. A water pump is a machine for raising water from 
a lower to a higher level, generally by the aid of atmospheric 
pressure. Three separate principles are employed in the 
working of pumps: the sucking, the lifting, and the 
forcing principle. Pumps are frequently named according 
as one or more of these principles are employed. 

Sucking and Lifting Pump. 

209. This pump consists of a 
cylindrical barrel A, at the lower 
extremity of which is attached a 
sucking-pipe JB, leading to a reser- 
voir. An air-tight piston C is work- 
ed up and down in the barrel by 
means of a lever E, attached to a 
piston-rod D. P represents a valve 
opening upwards, which, when the 
14 




^ 



Fig. 174 



314 MECHANICS. 

pump is at rest, closes by its own weight. This valve 
is called, from its position, the piston-valve. A second 
valve 67, also opening upwards, is placed at the junction of 
the pipe with the barrel. This is called the sleeping-valve. 
The space Xi!/, through which the piston can be moved up 
and down by the lever, is called the play of the piston. 

To explain the action of the pump, suppose the piston to 
be at the lowest limit of the play, and everything in a state 
of equilibrium. If the extremity of the lever E be 
depressed, and the piston consequently be raised, the air in 
the lower part of the barrel will be rarefied, and that in the 
pipe B will, by virtue of its greater tension, open the valve, 
and a portion of it will escape into the barrel. The air in 
the pipe, thus rarefied, will exert a less pressure upon the 
water in the reservoir than that of the external air, and, 
consequently, the water will rise in the pipe, until the tension 
of the internal air, plus the weight of the column of water 
raised, is equal to the tension of the external air ; the valve 
67 will then close by its own weight. 

If the piston be again depressed to the lowest limit, by 
means of the lever JEJ, the air in the lower part of the barrel 
will be compressed, its tension will become greater than that 
of the external air, the valve F will be forced open, and a 
portion of the air will escape. If the piston be raised once 
more, the water will, for the same reason as before, rise still 
higher in the pipe, and after a few double strokes of the 
piston, the air will be completely exhausted from beneath 
the piston, the water will pass through the piston valve, and 
finally escape at the spout P. 

The water is raised to the piston by the pressure of the 
air on the surface of the water in the reservoir ; hence, the 
piston should not be placed at a greater distance above the 
level of the water in the reservoir, than the height to which 
the pressure of the air will sustain a column of water. In 
fact, it should be placed a little lower than this limit. The 
specific gravity of mercury being about 13.5, the height of 
a column of water which will exactly counterbalance the 



HYDRAULIC AND PNEUMATIC MACHINES. 315 

pressure of the atmosphere, will be found by multiplying the 
height of the barometric column by 13^. 

At the level of the sea the average height of the baro- 
metric column is 2j feet ; hence, the theoretical height to 
which water can be raised by the principle of suction alone, 
is a little less than 34 feet. 

The water having passed through the piston valve, it may 
be raised to any height by the lifting principle, the only 
limitation being the strength of the pump and want of 
power. 

There are certain relations which must exist between the 
play of the piston and its height above the water in the 
reservoir, in order that the water may be raised to the 
piston ; for, if the play is too small, it will happen after a few 
strokes of the piston, that the air between the piston and 
the surface of the water will not be sufficiently compressed 
to open the piston valve ; when this state of affairs takes 
place, the water will cease to rise. 

To investigate the relation that must exist between the 
play and the height of the piston above the water. 

Denote the play of the piston, by p, the distance from the 
upper surface of the water in the reservoir to the highest 
position of the piston, by a, and the height at which the 
water ceases to rise in the pump, by x. The distance from 
the surface of the water in the pump to the highest position 
of the piston will then be equal to a — x, and the distance 
to the lowest position of the piston, will be a — p — x. 
Denote the height at which the atmospheric pressure will 
sustain a column of water in vacuum, by A, and the weight of 
a column of water, whose base is the cross-section of the 
pump, and whose altitude is 1, by w ; then will wh denote 
the pressure of the atmosphere exerted upwards through the 
water in the reservoir and pump. 

Xow, when the piston is at its lowest position, in order 
that it may not thrust open the piston valve and escape, the 
pressure of the confined air must be exactly equal to that 
of the external atmosphere; that is, equal to wh. When the 



316 MECHANICS. 

piston is at its highest position, the confined air will be rare* 
fied, the volume occupied being proportional to its height. 
Denoting the pressure of the rarefied air by wh' , we shall 
have from Mariotte's law, 

wh : wh' :: a — x : a — p — x. 

.-. wh' = wh — • 

a — x 

If the water does not rise when the piston is at its highest 
position, the pressure of the rarefied air, plus the weight of 
the column already raised, will be equal to the pressure of 
the external atmosphere ; or 

, a —p — x 

wh - h wx = wh. 

a — x 

Solving this equation with respect to x, we have, 



_ a ± -\/<tf — 4ph 

If we have, 

a 2 
4ph > a 2 ; or, p > — , 



the value of x will be imaginary, and there will be no point 
at which the water will cease to rise. Hence, the above 
inequality expresses the relation that must exist, in order 
that the pump may be effective. This condition expressed 
in words, gives the following rule : 

The pump will be effective, when the play of the piston is 
greater than the square of the distance from the surface of 
the water in the reservoir, to the highest position of the 
piston, divided by four times the height at which the pres- 
sure of the atmosphere will support a column of water in 
a vacuum. 

Let it be required to find the least allowable play of the 
piston, when the highest position of the piston is 16 feet 



HYDRAULIC AND PNEUMATIC MACHINES. 317 

above the water in the reservoir, and when the barometer 
stands at 28 inches. 
In this case, 

a = 16 ft., and h = 28 in. x 13£ = 378 in. = 3l£ ft. 

Hence, 

i? >f||ft.; or, p>2&ft. 

To find the quantity of work required to make a double 
stroke of the piston, after the water reaches the level of the 
spout. 

In depressing the piston, no force is required, except that 
necessary to overcome the inertia of the parts and the fric- 
tion. Neglecting these for the present, the quantity of 
work in the downward stroke, may be regarded as 0. In 
raising the piston, its upper surface will be pressed down- 
wards, by the pressure of the atmosphere wh, plus the weight 
of the column of water from the piston to the spout ; and it 
will be pressed upwards, by the pressure of the atmosphere, 
transmitted through the pump, minus the weight of a 
column of water, whose cross-section is equal to that of the 
barrel, and whose altitude is the distance from the piston to 
the surface of the water in the reservoir. If we subtract 
the latter pressure from the former, the difference will be 
the resultant downward pressure. This difference will be 
equal to the weight of a column of water, whose base is the 
cross-section of the barrel, and whose height is the distance 
of the spout above the reservoir. Denoting the height by 
H, the pressure will be equal to wH. The path through 
which the pressure is exerted during the ascent of the 
piston, is equal to the play of the piston, or^>. Denoting the 
quantity of work required, by §, we shall have, 

Q = wpH. 

But wp is the weight of a volume of water, whose base is 
the cross-section of the barrel, and whose altitude is the 
play of the piston. Hence, the value of Q is equal to the 



318 



MECHANICS. 



quantity of work necessary to raise this volume of watei 
from the level of the water in the reservoir to the spout. 
This volume is evidently equal to the volume actually 
delivered at each double stroke of the piston. Hence, the 
quantity of work expended in pumping with the sucking 
and lifting pump, all hurtful resistances being neglected, is 
equal to the quantity of work necessary to lift the amount 
of water, actually delivered, from the level of the water in 
the reservoir to the height of the spout. In addition to this 
work, a sufficient amount of power must be exerted, to 
overcome the hurtful resistances. The disadvantage of this 
pump, is the irregularity with which the force must act, 
being in depressing the piston, and a maximum in raising 
it. This is an important objection when machinery is eim 
ployed in pumping ; but it may be either partially or entirely 
overcome, by using two pumps, so arranged, that the piston 
of one shall ascend as that of the other descends. Another 
objection to the use of this kind of pump, is the irregularity 
of flow, the inertia of the column of water having to be 
overcome at each upward stroke. This, by creating shocks, 
consumes a portion of the force applied. 

Sucking and Forcing Pump. 

210. This pump consists of a cylindrical barrel A, with 
its attached sucking-pipe B, and 
sleeping-valve 6r, as in the pump 
just discussed. The piston G is 
solid, and is worked up and down 
in the barrel by means of a lever 
_F, attached to the piston-rod D. 
At the bottom of the barrel, a 
branch-pipe leads into an air-vessel 
K, through a second sleeping-valve 
F, which opens upwards, and closes 
by its own weight. A delivery- 
pipe H, enters the air-vessel at its 
top, and terminates near its bottom. 

To explain the action of this 




P 



K 



m 



Fig. 175. 



HYDRAULIC AND PNEUMATIC MACHINES. 319 

pump, suppose the piston C to be depressed to its lowest 
limit. Now, if the piston be raised to its highest position, 
the air in the barrel will be rarefied, its tension will be 
diminished, the air in the tube i?, will thrust open the valve, 
and a portion of it will escape into the barrel. The pres- 
sure of the external air will then force a column of water 
up the pipe B, until the tension of the rarefied ah*, plus the 
weight of the column of water raised, is equal to the tension 
of the external air. An equilibrium being produced, the 
valve G closes by its own weight. If, now, the piston be 
again depressed, the air in the barrel will be condensed, its 
tension will increase till it becomes greater than that of the 
external air, when the valve F will be thrust open, and a 
portion of it will escape through the delivery-pipe H. After 
a few double strokes of the piston, the water will rise 
through the valve 6r, and then, as the piston descends, it 
will be forced into the air-vessel, the air will be condensed 
in the upper part of the vessel, and, acting by its elastic 
force, will force a portion of the water up the delivery-pipe 
and out at the spout P. The object of the air-vessel is, to 
keep up a continued stream through the pipe 11, otherwise 
it would be necessary to overcome the inertia of the entire 
column of water in the pipe at every double stroke. The 
flow having commenced, at each double stroke, a volume of 
water will be delivered from the spout, equal to that of a 
cylinder whose base is the area of the piston, and whose 
altitude is the play of the piston. 

The same relative conditions between the parts should 
exist as in the sucking and lifting pump. 

To find the quantity of work consumed at each double 
stroke, after the flow has become regular, hurtful resistances 
being neglected : 

When the piston is descending, it is pressed downwards 
by the tension of the air on its upper surface, and upwards 
by the tension of the atmosphere, transmitted through the 
delivery-pipe, plus the weight of a column of water whose 
base is the area of the piston, and whose altitude is the 



320 MECHANICS. 

distance of the spout above the piston. This distance is 
variable during the stroke, but its mean vame is the distance 
of the middle of the play below the spout. The difference 
between these pressures is exerted upwards, and is equal to 
the weight of a colunm of water whose base is the area of 
the piston, and whose altitude is the distance from the 
middle of the play to the spout. The distance through 
which the force is exerted, is equal to the play of the piston. 
Denoting the quantity of work during the descending 
stroke, by Q' ; the weight of a column of water, having a 
base equal to the area of the piston, and a unit in altitude, 
by w ; and the height of the spout above the middle of the 
the play, by A', we shall have, 

Q' — wh' x p. 

When the piston is ascending, it is pressed downwards 
by the tension of the atmosphere on its upper surface, and 
upwards by the tension of the atmosphere, transmitted 
through the water in the reservoir and pump, minus the 
weight of a column of water whose base is the area of the 
piston, and whose altitude is the height of the piston above 
the reservoir. This height is variable, but its mean value 
is the height of the middle of the play above the water in 
the reservoir. The distance through which this force is 
exerted, is equal to the play of the piston. Denoting the 
quantity of work during the ascending stroke, by Q'\ and 
the height of the middle of the play above the reservoir, by 
A", we have, 

Q" = wh" x p. 

Denoting the entire quantity of work during a double stroke, 
by §, we have, 

Q = Q'+ Q" = wp{h! + A"). 

But wp is the weight of a volume of water, the area of 
whose base is that of the piston, and whose altitude is the 



HYDRAULIC AND PNEUMATIC MACHINES. 



321 



play of the piston ; that is, it is the weight of the volume 
delivered at the spout at each double stroke. 

The quantity h! + A", is the entire height of the spout 
above the level of the cistern. Hence, the quantity of work 
expended, is equal to that required to raise the entire volume 
delivered, from the level of the water in the reservoir to the 
height of the spout. To this must be added the work 
necessary to overcome the hurtful resistances, such as fric- 
tion, &c. 

If h' = h'\ we shall have, Q' = Q" ; that is, the quan- 
tity of work during the ascending stroke, will be equal to 
that during the descending stroke. Hence, the work of the 
motor will be more nearly uniform, when the middle of the 
play of the piston is at equal distances from the reservoir 
and spout. 

Fire Engine. 

211. The fire engine is essentially a double sucking and 
forcing pump, the two piston rods being so connected, that 
when one piston ascends the other descends. The sucking 
and delivery pipes are made of some flexible material, gen- 
erally of leather, and are attached to the machine by means 
of metallic screw joints. 

The figure exhibits a cross-section of the essential part of 
a Fire Engine. 

A A' are the two barrels, G C the two pistons, con- 
nected by the rods, D D , 
with the lever, E E'. B J! 
is the sucking pipe, termi- 
nating in a box froru 
which the water may en- 
ter either barrel through 
the valves, G G'. K is 
the air vessel, common to 
both pumps, and com- 
municating with them by 
the valves F F '. H is 

the delivery pipe. 
14* 




Fig. 17& 



322 



MECHANICS. 



The instrument is mounted on wheels for convenience of 
transportation. The lever E E' is worked by means of 
rods at right angles to the lever, so arranged that several 
men can apply their strength in working the pump. The 
action of the pump differs in no respect from that of the 
forcing pump ; but when the instrument is worked vigor- 
ously, there is more water forced into the air vessel, the 
tension of the air is very much augmented, and its elastic 
force, thus brought into play, propels the water to a consider- 
able distance from the mouth of the delivery pipe. It is 
this capacity of throwing a jet of water to a great distance, 
that gives to the engine its value in extinguishing fires. 

A pump entirely similar to the fire engine in its construc- 
tion, is often used under the name of the double action forc- 
ing pump for raising water for other purposes. 

The Rotary Pump. 

212. The rotary pump is a modification of the sucking 
and forcing pump. Its construction will be best understood 
from the drawing, which represents a vertical section through 
the axis of the sucking-pipe, and at right angles to axis of 
the rotary portion of the pump. 

A represents an annular ring of metal, which may be 
made to revolve about its axis 
0. D D is a second ring of 
metal, concentric with the first, 
and forming with it an inter- 
mediate annular space. This 
space communicates with the 
sucking-pipe E, and the de- 
livery pipe L. Four radial 
paddles (7, are disposed so as 
to slide backwards and for- 
wards through suitable open- 
ings, which are made in the 

ring A, and which are moved around with it. G is a solid 
guide, firmly fastened to the end of the cylinder enclosing 




Fig. 177. 



HYDRAULIC AND PNEUMATIC MACHINES. 323 

the rotary apparatus, and cut as represented in the figure. 
E E are two springs, attached to the ring Z>, and. acting by 
their elastic force, to press the paddles firmly against the 
guide. These springs are of such dimensions as not to 
impede the flow of the water from the pipe E, and into the 
pipe L. 

When the axis is made to revolve, each paddle, as it 
reaches and passes the partition II, is pressed against the 
guide, but, as it moves on, it is forced, by the form of the 
guide, against the outer wall D. The paddle then drives 
the air in front of it, around, in the direction of the arrow- 
head, and finally expels it through the pipe L. The air 
behind the paddle is rarefied, and the pressure of the exter- 
nal air forces a column of water up the pipe. As the paddle 
approaches the opening to the pipe _Z, the paddle is pressed 
back by the spring E, against the guide, and an outlet into 
the ascending pipe I, is thus provided. After a few revo- 
lutions, the air is entirely exhausted from the pipe K. The 
water enters the channel C C, and is forced up the pipe X, 
from which it escapes by a spout at the top. The quantity 
of work expended in raising a volume of water to the 
spout, by this pump, is equal to that required to lift it 
through the distance from the level of the water in the cis- 
tern to the spout. This may be shown in the same manner 
as was explained under the head of the sucking and forcing- 
pump. To this quantity of work, must be added the work 
necessary to overcome the hurtful resistances, as fric- 
tion, &c. 

This pump is well adapted to machine pumping, the work 
being very nearly uniform. 

A machine, entirely similar to the rotary pump, might be 
constructed for exhausting foul air from mines ; or, by re- 
versing the direction of rotation, it might be made to force 
a supply of fresh air to the bottom of deep mines. 

Besides the pumps already described, a great variety 
of others have been invented and used. All, however, 



324 



MECHANICS. 



fMf 



E 



depend upon some modification of the principles that have 
just been discussed. 

The Hydrostatic Press. 

213. The hydrostatic press is a machine for exerting 
great pressure through small spaces. It is much used in 
compressing seeds to obtain oil, in packing hay and bales of 
goods, also in raising great weights. Its construction, though 
requiring the use of a sucking-pump, depends upon the prin- 
ciple of equal pressures (Art. 154). 

It consists essentially of two vertical cylinders, A and B, 
each provided with a solid pis- 
ton. The cylinders communi- 
cate by means of a pipe (7, 
whose entrance to the larger 
cylinder is closed by a sleeping 
valve F. The smaller cylinder 
communicates with the reser- 
voir of water K, by a sucking- 
pipe IT, whose upper extremity 
is closed by the sleeping-valve D. 
The smaller piston B, is worked up and down by the lever 
G. By working the lever G, up and down, the water is 
raised from the reservoir and forced into the larger cylinder 
A ; and when the space below the piston F is filled, a force 
of compression is exerted upwards, which is as many times 
greater than that applied to the piston B, as the area of 
j^is greater than B (Art. 154). This force may be util- 
ized in compressing a body X, placed between the piston 
and the frame of the press. 

Denote the area of the larger piston by J°, of the smaller, 
by p, the pressure applied to B, by /, and that exerted at 
F, by F; we shall have, 



□ 



H 



Fig. 178. 



F:f::P:p, 



P 



If we denote the longer arm of the lever G, by X, and 



HYDRAULIC AND PNEUMATIC MACHINES. 325 

the shorter arm, by I, and represent the force applied at the 
extremity of the longer arm, by K, we shall have from the 
principle of the lever (Art. 78), 

K:f.:l:L, ,. f=^. 

Substituting this value of /"above, we have, 

PKL 



F 



pi 



To illustrate, let the area of the larger piston be 100 
square inches, that of the smaller piston 1 square inch ; sup- 
pose the longer arm of the lever to be 30 inches, and the 
shorter arm to be 2 inches, and a force of 100 pounds to be 
applied at the end of the longer arm of the lever ; to find 
the pressure exerted upon F. 

From the conditions, 

P =. 100, K - 100, L = 30, p = 1, and I =: 2. 

Hence, 

^ 100 X 100 x 30 1e „ rt ^ lv 
F= = 150000 lbs. 

We have not taken into account the hurtful resistances, 
hence, the total pressure of 150000 pounds must be some- 
what diminished. 

The volume of water forced from the smaller to the larger 
piston, during a single descent of the piston F\ will occupy 
in the two cylinders, spaces whose heights are inversely as 
the areas of the pistons. Hence, the path, over which f ia 
exerted, is to the path over which F is exerted, as P is to 
p. Or, denoting these paths by s and $, we have, 

s : S : : P : p; 

or, since P : p : : F : f, we shall have, 

s : S:: F:f, .-. fs = FS. 



OZb MECHANIC?. 

That is, the quantities of work of the power and resistance 
are equal, a principle which holds good in all machines. 

EXAMPLES . 

1. The cross-section of a sucking and forcing pump is 6 
square feet, the play of the piston 3 feet, and the height of 
the spout, above the level of the reservoir, 50 feet. What 
must be the effective horse power of an engine which can 
impart 30 double strokes per minute, hurtful resistances 
being neglected ? 

SOLUTION. 

The number of units of work required to be performed 
each minute, is equal to 



Hence, 

m 56251 — 1 _9_3_ An* 

" J — 33000 — x 1 3 2 * -£L/*o. 

2. In a hydrostatic press, the areas of the two pistons are, 
respectively, 2 and 400 square inches, and the two arms of 
the lever are, respectively, 1 and 20 inches. Required the 
pressure on the larger piston for each pound of pressure 
applied to the longer arm of the lever ? Ans. 4000 lbs. 

3. The areas of the two pistons of a hydrostatic press 
are, respectively, equal to 3 and 300 square inches, and the 
shorter arm of the lever is one inch. What must be the 
length of the longer arm, that a force of 1 lb. may produce 
a pressure of 1000 lbs. Ans. 10 inches. 

The Siphon. 

214. The siphon is a bent tube, used for transferring a 
liquid from a higher to a lower level, over an in- 
termediate elevation The siphon consists of two 
branches, AB and BC, of which the outer one 
is the longer. To use the instrument, the tube 
is filled with the liquid in any manner, the end of §H 
the longer branch being stopped with the finger I 

or a stop-cock, in which case, the pressure of the Jf 

atmosphere will prevent the liquid from escaping Fig . 17 9, 



HYDRAULIC AND PNEUMATIC MACHINES. 327 

at the other end. The instrument is then inverted, 
the end G being submerged in the liquid, and the stop 
removed from A. The liquid will begin to flow through 
the tube, and the flow will continue till the level of the 
liquid in the reservoir reaches that of the mouth of the 
tube C. 

To find the velocity with which water will issue from the 
siphon, let us consider an infinitely small layer at the orifice 
A. This layer will be pressed downwards, by the tension 
of the atmosphere exerted on the surface of the reservoir, 
diminished by the weight of the water in the branch BD, 
and increased by the weight of the water in the branch 
BA. It will be pressed upwards by the tension of the 
atmosphere acting directly upon the layer. The difference 
of these forces, is the weight of the water in the portion of 
the tube DA, and the velocity of the stratum will be due 
to that weight. Denoting the vertical height of DA, by h, 
we shallhave, for the velocity (Art. 173), 



This is the theoretical velocity, but it is never quite 
realized in practice, on account of resistances, which have 
been neglected in the preceding investigation. 

The siphon may be filled by applying the mouth to the 
end A, and exhausting the air by suction. The 
tension of the atmosphere, on the upper surface 
of the reservoir, will press the water up the tube, 
and fill it, after which the flow will go on as 
before. Sometimes, a sucking-tube AD, is in- 
serted near the opening A, and rising nearly to 
the bend of the siphon. In this case, the opening 
A, is closed, and the air exhausted through the 
sucking-tube AD, after which the flow goes on as before. 

The Wurtemburg Siphon. 
215. In the Wurtemburg siphon, the ends of the tube are 



328 



MECHANICS. 



Ft 



w 

181. 



bent twice, at right-angles, as shown in the figure. 
The advantage of this arrangement is, that the 
tube, once filled, remains so, as long as the plane 
of its axis is kept vertical. The siphon may be 
lifted out and replaced at pleasure, thereby 
stopping the flow at will. 

It is to be observed that the siphon is only effectual when 
the distance from the highest point of the tube to the level 
of the water in the reservoir is less than the height at which 
the atmospheric pressure will sustain a column of water in 
a vacuum. This will, in general, be less than 34 feet. 



The Intermitting Siphon. 

216. The intermitting siphon is represented in the 
figure. AB is a curved tube issuing 
from the bottom of a reservoir. The 
reservoir is supplied with water by a 
tube E, having a smaller bore than 
that of the siphon. To explain its 
action, suppose the reservoir at first 
to be empty, and the tube E to be 
opened; as soon as the reservoir is 
filled to the level of CD, the water 
will begin to flow from the opening 

B, and the flow once commenced, will continue till the 
level of the reservoir is again reduced to the level C'D\ 
drawn through the opening A. The flow will then cease 
till the cistern is again filled to (7Z>, and so on as before. 




Intermitting Springs. 

217. Let A represent a subterranean cavity, communi- 
cating with the surface of the earth by 
a channel ABC, bent like a siphon. 
Suppose the reservoir to be fed by 
percolation through the crevices, or 
by a small channel I). When the Fig. 18& 




HYDRAULIC AND PNEUMATIC MACHINES. 329 

water iii the reservoir rises to the height of the horizontal 
plane J3D, the flow will commence at (7, and, if the chan- 
nel is sufliciently large, the flow will continue till the water 
is reduced to the level plane drawn through C. An inter- 
mission of flow will occur till the reservoir is again filled, 
and so on, intermittingly. This phenomena has been observed 
at various places. 

Siphon of Constant Flow. 

218. We have seen that the velocity of efflux depends 
upon the height of the water in the reservoir above the 
external opening of the siphon. When the water is drawn 
off from the reservoir, the upper surface sinks, this height 
diminishes, and, consequently, the velocity continually 
diminishes. 

If, however, the shorter branch CD, of the tube, be 
inserted through a piece of cork large enough to float the 
siphon, the instrument will sink as the upper surface is 
depressed, the height of DA will remain the same, and, 
consequently, the flow will be uniform till the bend of the 
siphon comes in contact with the upper edge of the reservoir. 
By suitably adjusting the siphon in the cork, the velocity 
of efflux can be increased or decreased within certain limits. 
In this manner, any desired quantity of the fluid can be 
drawn off in a given time. 

The siphon is used in the arts, for decanting liquids, when 
it is desirable not to stir the sediment at the bottom of a 
vessel. It is also employed to draw a portion of a liquid 
from the interior of a vessel when that liquid is overlaid by 
one of less specific gravity. 

The Hydraulic Ram. 

219. The hydraulic ram is a machine for raising watei 
by means of shocks caused by the sudden stoppages of a 
stream of water. 

The instrument consists of a reservoir D, which is sup- 
plied with water by an inclined pipe A ; on the upper surface 




EE~~ *~~^S 



B 



330 MECHANICS. 

of the reservoir, is an orifice which may be closed by 
a spherical valve D ; this valve, 
when not pressed against the 
opening, rests in a metallic 
framework immediately below 
the orifice ; G is an air-vessel 
communicating with the reser- ~ 
voir by an orifice JF] which is 
fitted with a spherical valve E\ 
this valve closes the orifice F, 
except when forced upwards, 

in which case its motion is restrained by a metallic frame 
work or cage ; H represents a delivery-pipe entering the 
air-vessel at its upper part, and terminating near the bot- 
tom. At P is a small valve, opening inwards, to supply 
the loss of air in the air-vessel, arising from absorption by 
the water in passing through the air vessel. 

To explain the action of the instrument, suppose, at first, 
that it is empty, and all the parts in equilibrium. If a cur- 
rent of water be admitted to the reservoir, through the in- 
clined pipe A, the reservoir will soon be filled, and com- 
mence rushing out at the orifice C. The impulse of the 
water will force the spherical valve D, upwards, closing the 
opening ; the velocity of the water in the reservoir will be 
suddenly checked ; the reaction will force open the valve 
jEJ and a portion of the water will enter the air-chamber G. 
The force of the shock having been expended, the spherical 
valves will both fall by their own weight ; a second shock 
will take place, as before ; an additional quantity of water 
will be forced into the air-vessel, and so on, indefinitely. 
As the water is forced up into the air-vessel, the air becomes 
compressed ; and acting by its elastic force, it urges a stream 
of water up the pipe H. The shocks occur in rapid succes- 
sion, and, at each shock, a quantity of water is forced into 
the air-chamber, and thus a constant stream is kept up. 
To explain the use of the valvt P, it maybe remarked that 
water absorbs more air under a great pressure, than under 



HYDRAULIC AND PNEUMATIC MACHINES. 331 

a smaller one. Hence, as it passes through the air-chamber, 
a portion of the air contained is taken up by the water and 
carried out through the pipe H. But each time that the 
valve D falls, there is a tendency to produce a vacuum 
in the upper part of the reservoir, in consequence of the 
rush of the fluid to escape through the opening. The pres- 
sure of the external air then forces the valve P open, a 
small portion of air enters, and is afterwards forced up with 
the water into the vessel 6?, to keep up the supply. 

The hydraulic ram is only used where it is required to 
raise small quantities of water, such as for the supply of a 
house, or garden. Only a small fraction of the amount of 
fluid which enters the supply-pipe actually passes out 
through the delivery-pipe; but, if the head of water is 
pretty large, the column may be raised to a great height. 
Water is often raised, hi this manner, to the highest points 
of lofty buildings. 

Sometimes, an additional air-vessel is introduced over the 
valve E, for the purpose of deadening the shock of the 
valve in its play up and down. 

Archimedes' Screw. 

220. This machine is intended for raising water through 
small heights, and consists, in its simplest form, of a tube 
wound spirally around a cylinder. This cylinder is mounted 
so that its axis is oblique to the horizon, the lower end dip- 
ping into the reservoir. When the cylinder is turned on its 
axis, by a crank attached to its upper extremity, the lower 
end of the tube describes a circumference of a circle, whose 
plane is perpendicular to the axis. When the mouth of the 
tube comes to the level of the axis and begins to ascend, 
there will be a certain quantity of water in the tube, which will 
flow so as to occupy the lowest part of the spire ; and, if the 
cylinder is properly inclined to the horizon, this flow will be 
towards the upper end of the tube. At each revolution, an 
additional quantity of water will enter the tube, and that 
already in the tube will be forced, or raised, higher and 



33: 



MECHANICS. 




Fig. 185. 



higher, till, at last, it will flow from the orifice at the uppej 
end of the spiral tube. 

The Chain Pump. 

221. The chain pump is an instrument for raising watei 
through small elevations. It consists 

of an endless chain passing over two 
wheels, A and .2?, having their axes 
horizontal, the one being below the 
surface of the water, and the other 
above the spout 'of the pump. At- 
tached to this chain, and at right 
angles to it, are a system of circular 
disks, just fitting the tube CD. If 
the cylinder A be turned in the di- 
rection of the arrow-head, the buckets 
or disks will rise through the tube 
CD, carrying the water in the tube before them, until it 
reaches the spout (7, and escapes. The buckets thus emptied 
return through the air to the reservoir, and so on perpetually. 
One great objection to this machine is, the difficulty of 
making the buckets fit the tube of the pump. Hence there 
is a constant leakage, requiring a great additional expend- 
iture of force. 

Sometimes, instead of having the body of the pump ver- 
tical, it is inclined ; in which case it does not differ much 
in principle from the wheel with flat buckets, that has been 
used for raising water. 

The Air Pump. 

222. The air pump is a machine for rarefying the air in 
a closed space. 

It consists of a cylindrical 
barrel A, in which a piston 
B, fitting air-tight, is work- 
ed up and down by a lever 
(7, attached to a piston-rod 
D. The barrel communi- 
cates with an air-tight ves- 




Fig. 186. 



HYDRAULIC AND PNEUMATIC MACHINES. 333 

sel E, called a receiver, by means of a narrow pipe. The 
receiver, which is usually of glass, is ground so as to fit air- 
tight upon a smooth bed-plate KK. The joint between the 
receiver and plate may be rendered more perfectly air-tight, 
by rubbing it with a little oil. A stop-cock H, of a peculiar 
construction, permits communication to be made at pleasure 
between the barrel and receiver, or between the barrel and 
the external air. When the stop-cock is turned in a partic- 
ular direction, the barrel and receiver are made to commu- 
nicate ; but on turning it through 90 degrees, the communi- 
cation with the receiver is cut off, and a communication is 
opened between the barrel and the external air. Instead of 
the stop-cock, valves are often used, which are either opened 
and closed by the elastic force of the air, or by the force 
that works the pump. The communicating pipe should be 
exceedingly small, and the piston B should, when at its low- 
est point, tit accurately to the bottom of the barrel. 

To explain the action of the air pump, suppose the piston 
to be depressed to its lowest position. The stop-cock II, is 
turned so as to open a communication between the barrel 
and receiver, and the piston is raised to its highest point by 
a force applied to the lever C. The air which before occu- 
pied the receiver and pipe, will expand so as to fill the bar- 
rel, receiver, asid pipe. The stop-cock is then turned so as to 
cut off communication between the barrel and receiver, and 
open the barrel to the external air, and the piston again de- 
pressed to its lowest position. The rarefied air in the barrel 
is expelled into the external air by the depression of the 
piston. The air in the receiver is now more rarefied than at 
the beginning, and by a continued repetition of the process 
just described, any degree of rarefaction may be attained. 

To measure the degree of rarefaction of the air in the 
receiver, a siphon-gauge may be used, or a glass tube, 30 
inches long, may be made to communicate at its upper 
extremity with the receiver, whilst its lower extremity dips 
into a cistern of mercury. As the air is rarefied in the 
receiver, the pressure on the mercury in the tube becomes 



334 MECHANICS. 

less than that on the surface of the mercury in the cistern , 
and the mercury rises in the tube. The tension of the air 
in the receiver will be given by the difference between the 
height of the barometric column and that of the mercury 
in the tube. 

To investigate a formula for computing the tension of the 
air in the receiver, after any number of double strokes, let 
us denote the capacity of the receiver in cubic feet, by r, 
that of the connecting-pipe, by p, and the space between 
the bottom of the barrel and the highest position of the 
piston, by b. Denote the original tension of the air, by t ; 
its tension after the first upward stroke of the piston, by t' ; 
after the second, third, . . . n th , upward strokes, by 
t, t", . . . t n '. 

The air which originally occupied the receiver and pipe, 
fills the receiver, pipe, and barrel, after the first upward 
stroke ; according to Mariotte's law, its tension in the two 
cases varies inversely as the volumes occupied ; hence, 



t • t' : : p + r -4- b : p + r, .-. t' = t 



p-\- r + b 



In like manner, we shall have, after the second upward 
stroke, 

t':t"::p + r + l>:p + r, .: t" = t'-t+l-. 

Substituting for t r its value, deduced from the preceding 
equation, we have, 

t" -t( p + r V 

1 -^p + b + r) 

In like manner, we find, 

,,„ t ( P + r <\ 



HYDRAULIC AND PNEUMATIC MACHINES. 335 

and, in general, 

/«' — t.l 



\x> 4- o 4-r / 



If the pipe is exceedingly small, its capacity may be 
neglected in comparison with that of the receiver, and we 
shall then have, 



\b + r) 



Let it be required, for example, to determine the tensiou 
of the air after 5 upward strokes, when the capacity of the 
barrel is one-third that of the receiver. 

T 

In this case, ■= =r -|, and n = 5, whence, 

b + r 4 

*« _ / 24 3 , 

Hence, the tension is less than a fourth part of that the 
external air. 

Instead of the receiver, the pipe may be connected by a 
screw-joint with any closed vessel, as a hollow globe or glass 
flask. In this case, by reversing the direction of the stop- 
cock, in the up and down motion of the piston, the in- 
strument may be used as a condenser. When so used, the 
tension, after n downward strokes of the piston, is given by 
the formula, 

b + r 



t« 



(^-)- 



Taking the same case as that before considered, with the 
e3 ception that the instrument is used as a condenser instead 
of a rarefier, we have, after 5 downward strokes, 



fv / 102 4 



That is, the tension is more than four times that of the 
external air. 




336 MECHANICS. 

When the pump is used for condensing air, it is called a 
condenser. 

Artificial Fountains. 

223. An artificial fountain is an instrument by means of 
which a liquid is forced upwards in the form of a jet, by 
the tension of condensed air. The simplest form of an arti- 
ficial fountain is called Hero's ball. 

Hero's Ball. 

224. This instrument consists of a hollow globe A, into 
the top of which is inserted a vertical tube J?, 

reaching nearly to the bottom of the globe. 
This tube is provided with a stop-cock O, by 
means of which it may be closed, or opened to 
the external air, at pleasure. A second tube 
J9, enters the globe near the top, which is also 
provided with a stop-cock E. 

To use the instrument, close the stop-cock (7, Fig. 187. 

and fill the lower portion of the globe with 
water through the tube D ; then attach the tube D to a 
condenser, and pump air into the upper part of the globe, 
and confine it there by closing the stop-cock E If, now, the 
stop-cock G be opened, the pressure of the confined air on 
the surface of the water in the globe, will force a jet up 
through the tube B. This jet will rise to a greater or less 
height, according to the greater or less quantity of air that 
was forced into the globe. The water will continue to flow 
through the tube as long as the tension of the confined air 
is greater than that of the external atmosphere, or else till 
the level of the water in the globe reaches the lower end 
of the tube. 

Instead of using the condenser, air may be introduced by 
blowing with the mouth through the tube D, and then con- 
fined as before, by turning the stop-cock E. 

The principle of Hero's ball is the same as that of the air- 
chamber in the forcing pump and fire-engine, already ex 
plained. 



HYDRAULIC AND PNEUMATIC MACHINES. 



337 



r a 



A 



¥ 



Fig. 188 



Hero's Fountain. 

225. Hero's fountain is constructed on the same prin- 
ciple as Hero's ball, except that the compression of the air 
is effected by the weight of a column of water, instead of by 
aid of a condenser 

A represents a cistern, similar to Hero's ball, with a tube 
J?, extending nearly to the bottom of the cis- 
tern. C is a second cistern placed at some & 
distance below A. This cistern is connected 
with a basin D, by a bent tube E, and also 
with the upper part of the cistern A, by a 
tube F. When the fountain is to be used, 
the cistern A is nearly filled with water, 
the cistern C being empty. A quantity of 
water is then poured into the basin 7>, which, 
acting by its weight, sinks into the cistern C, 
compressing the air in the upper portion of it 
into a smaller space, thus increasing its tension. 
This increase of tension acting on the surface 
of the water in A, forces a jet through the tube JB, which 
rises to a greater or less height according to the greater or 
less increase of the atmospheric tension. The flow will con- 
tinue till the level of the water in A, reaches the bottom of 
the tube JB. The measure of the compressing force on a 
unit of surface of the water in C, is the weight of a column 
of water, whose base is a square unit, and whose altitude is 
the difference of level between the water in D and C. 

If Hero's ball be partially filled with water and placed 
under the receiver of an air pump, the water will be ob- 
served to rise in the tube, forming a fountain, as the air in 
the receiver is exhausted. The principle is the same as 
before, an excess of pressure on the water within the globe 
over that without. In both cases, the flow is resisted by the 
tension of the air without, and is urged on by the tension 

within. 

Wine-Taster and Dropping-Bottle. 

226. The wine-taster is used to bring up a small por- 



338 MECHANICS. 

tion of wine or other liquid, from a cask. It 
consists of a tube, open at the top, and terminat- 
ing below in a very narrow tube, also open. When 
it is to be used, it is inserted to any depth in the 
liquid, which will rise in the tube to the level of 
the upper surface of that liquid. The finger is 
then placed so as to close the upper orifice of 
the tube, and the instrument is raised out of the ** l 
cask. A portion of the fluid escapes from the lower orifice, 
until the pressure of the rarefied air in the tube, plus the 
weight of a column of liquid, whose cross-section is that of 
the tube, and whose altitude is that of the column of fluid 
retained, is just equal to the pressure of the external air. 
If the tube be placed over a tumbler, and the finger re- 
moved from the upper orifice, the fluid brought up will 
escape into the tumbler. 

If the lower orifice is very small, a few drops may be 
allowed to escape, by taking off the finger and immediately 
replacing it. The instrument then constitutes the dropping 

tube. 

The Atmospheric Inkstand. 

227. The atmospheric inkstand consists of a cylinder 
A, which communicates by a tube with a 
second cylinder B. A piston C, is moved 
up and down in A, by means of a screw D. 
Suppose the spaces A and _Z?, to be filled 
with ink. If the piston C is raised, the 
pressure of the external air forces the ink to 
follow it, and the part B is emptied. If the Fig. iso 

operation be reversed, and the piston 
depressed, the ink is again forced into the space B. This 
operation may be repeated at pleasure. 




APPENDIX. 



The following notes contain elementary demonstrations 
of those principles, which in the body of the work are 
proved by means of the Calculus. 

Note on Articles 64—70 ; pp. 72-76. 

These articles may be omitted without at all impairing 
the unity of the subject, the preceding principles being suf- 
ficient to find the centre of gravity of all bodies, approxima- 
te ely. 

Note on Articles 112—114; pp. 143—148. 

The principal formulas in these articles may be deduced 
as follows : 

112. By definition, a body moves uniformly when it 
passes over equal spaces in equal times ; now if it passes 
over a space v in one second, it will pass over t times that 
space in t seconds ; that is, it will pass over a space vt. If 
we suppose it to have passed over a space s' before the com- 
mencement of the time t, we shall have for the entire space 
passed over, and which may be denoted by s, 

s — vt -f s' . . . . -. (58.) 

This equation corresponds to Equation (58) of the text. 

113. The formulas of Article 113 may be omitted with- 
out impairing the unity of the course. They are only of 
use in Higher Mechanics, where the employment of the Cal- 
culus is a necessity. 



3tt0 MECHANICS. 

114. Uniformly varied motion, is that in which the 
velocity increases or diminishes uniformly. In the former 
case the motion is accelerated, in the latter it is retarded. 
In "both cases the moving force is constant. 

Denote the moving force by/*, the mass moved being the 
unit of mass. 

According to Art. 24, the measure of the force is the ve- 
locity impressed in a unit of time, that is, in 1 second. Now 
from the principal of inertia, Art. 1 8, it follows that a force 
will produce the same general effect upon a body, whether 
it finds the body at rest or in motion. Hence, the velocit) 
impressed in any second of time is constant ; that is, if the 
Telocity impressed in one second of time is f in t seconds 
it will be t times f or ft. Denoting the velocity by v, 
we shall have, 

V =ft (69.) 

If the body has a velocity v' at the beginning of the time 
t, this velocity is called the initial velocity. Adding this to 
the velocity imparted during the time t, we have, 

v = v' + ft (67.) 

With respect to the space passed over, it may be re- 
marked that the velocity increases uniformly; hence the 
space passed over in any time, is the same that it would 
have passed over in the same time, had it moved uniformly 
during that time with its mean or average velocity. Now, 
if a body start from a state of rest, its velocity at starting is 
0, and at the end of the time t it is ft, Equation (69) ; the 
average or menu of these is \ft. But the space described 
in the time t, when the body moves with the uniform ve- 
locity \ft, is (Equation 55) equal to \ft X t ; denoting 
the space by s, we have, 

s = ift 2 (70.) 

If in Equation (7 ), we make t — 1, we have, 
s - \f\ or, / = 2s ; 



APPENDIX. 341 

that is, if a body moves from a state of rest, the space de- 
scribed in the first second of time, is equal to half the 
measure of the accelerating force ; or, the acceleration is 
measured by twice the space passed over in one second of 
time. 

If we suppose that a body starts from rest before the be- 
ginning of the time t, so as to pass over a space s' before 
the beginning of t, it will during that time have acquired 
some velocity, which we may denote by v'. The space 
reckoned from the origin of spaces up to the position of the 
body at the end of the time t, is made up of three parts ; 
first, the space s', called the initial space ; second, a space 
due to the velocity v' during the time t, which is measured 
by v't; third, a space due to the action of the incessant force 
during the time £, which will (Equation 70) be equal to 
ift 2 . Adding these together, we have finally, 

s = s'+ v't + ift 2 . . • (68.) 

If, in Equations (67) and (68), we suppose /"to be essen- 
tially positive, the motion will be accelerated; if we suppose 
it to be essentially negative, the motion will be retarded, 
and these equations become 

v = v' -ft (71.) 

s = s' + v't - Ift 2 . . . . ( 72.) 

Note on Article 121, pp. 163—164. 

The formula deduced in the first part of this article is 
needed in the investigations of Acoustics and Optics, and 
can only be found by the Calculus. This part of the article 
may be omitted without impairing the unity of the course. 

Note on Article 123, pp. 166—168. 

This article, up to the end of Equation (95), may be re- 
placed by the following deMonstration : 



U2 



MECHANICS. 



The simple pendulum. 

123. A pendulum is a heavy body suspended from a 
horizontal axis about which it is free to vibrate. 

In order to investigate the circumstances of vibration, let 
us first consider the hypothetical case of a single material 
point, vibrating about an axis to which it is attached by a 
rod destitute of weight. Such a pendulum is called a 
simple pendulum. The laws of vibration in this case will 
"be identical with those explained in Art. 120, the arc ABO 
being an arc of a circle. 

Let AB G be the arc through 
which the vibration takes place, 
and denote its radius DA, by I. 
The 'angle ADO is called the 
amplitude of vibration ; half of 
this angle, ADB, is called the 
angle of deviation. 

If the point starts from rest at 
A, it will, on reaching any point 
H, have a velocity v, due to the 
height EK, denoted by A, (Art. 
120). Hence, 




(92.) 



Let us suppose that the angle of deviation is so small, that 
the chords of the arcs AB and JIB, may be considered 
equal to the arcs themselves. We shall have (Davies' Le- 
gendre, Bk. IV., Prop. XXIIL, Cor.), 



AB 2 = 21 x EB, and ~HB' A = 21 x KB, 
whence, by subtraction, 

AB 2 - HB 2 = 2l{EB - KB) = 21 x h. 



APPENDIX. 343 

Denoting AB by a, and HB by x, and solving the 
last equation, we have, 

a 2 - x* 
h = 



21 
Substituting this value of h in (92) it becomes, 



= \/f^ 2 -* 2 ) • • • • («•) 



Now let us develop the arc ABO into a straight line 
A ' B ' C\ and suppose a material point to start from A' at 
the same time that the pendulum starts from A, and to 
vibrate back and forth upon A'B' C with the same veloci- 
ties as the pendulum; then, when the pendulum is at any 
point H, this material point will be at the corresponding 
point H\ and the times of vibration of the two will be 
exactly the same. 

To find the time of vibration along the line A'B'C, de- 
scribe upon it a semi-circle A' MG\ and suppose a third 
material point to start from A' at the same time as the 
second, and to move uniformly around the arc with a ve- 
locity equal to a\J j • Then will the time required for 

this particle to reach C be equal to the space divided by 
the velocity (Art. 112). Denoting this time by % and re- 
membering that A'B' = «, we shall have, 



9 v 

Make JTB' = x, and draw II' M perpendicular to A'C, 
and at M decompose the velocity of the third particle 
MT into two components MN and MQ, respectively par 
allel and perpendicular to A'C. 



344 MECHANICS. 

We shall have for the horizontal component MN\ 
3IJST = MT cos TMN. 

But, MT = «\/y, and because MT and JJfiV are re- 
spectively perpendicular to J3M and JIM, we have, 
cos rjlfiV = cos B'MH' = ^!~, But B'M = a. 

and iTJf = -/a 2 —"a? ; hence, cos CTfiV = 



a 

Substituting these values in Equation (&), we have for 

the horizontal velocity, 



V> 



MN = yj^(a 2 -x 2 ), 

which is the same value as that obtained for v in Equa- 
tion (a). Hence, we infer that the velocity of the third 
material point in the direction of A' C is always equal to 
that of the second point, consequently the times required 
to pass from A' to C must be equal ; that is, the time 
of vibration of the second point, and consequently of the 

pendulum, must be *\J - • Denoting this time by t, we 
have, 

t = *\/~ . . ". . . (95.) 



Note on Article 131, pp. 182—186. 

This article may be omitted without impairing the unity 
of the course. The results may be assumed if needed. 
They can only be deduced by the Calculus by demon- 
strations too tedious for an Elementary Course. 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



MATHEMATICS. 



DAVIES'S COMPLETE SERIES. 

ARITHMETIC. 

Davies' Primary Arithmetic. 

Davies' Intellectual Arithmetic. 

Davies' Elements of Written Arithmetic. 

Davies' Practical Arithmetic. 

Davies' University Arithmetic. 

TWO-BOOK SERIES. 

First Book in Arithmetic, Primary and Mental. 
Complete Arithmetic. 

ALGEBRA. 
Davies' New Elementary Algebra. 
Davies' University Algebra. 
Davies' New Bourdon's Algebra. 

GEOMETRY. 

Davies' Elementary Geometry and Trigonometry. 

Davies' Legendre's Geometry. 

Davies' Analytical Geometry and Calculus. 

Davies' Descriptive Geometry. 

Davies' New Calculus. 

MENSURATION. 

Davies' Practical Mathematics and Mensuration, 

Davies' Elements of Surveying. 

Davies' Shades, Shadows, and Perspective. 

MATHEMATICAL SCIENCE. 
Davies' Grammar of Arithmetic. 
Davies' Outlines of Mathematical Science. 
Davies' Nature and Utility of Mathematics. 
Davies' Metric System. 
Davies & Peck's Dictionary of Mathematics. 

20 




THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 

NATURAL SCIENCE— Continued. 

THE NEW SURVEYING. 

Van Amringe's Davies' Surveying. 

By Charles Davies, LL.D., author of a Full Course uf Mathematics. Revised by J 
Howard Van Amringe, KM., Ph.D., Professoi of Mathematics hi Columbia College 
56tf pages. 8vo. Full sheep. 

Davies' Surveying originally appeared as a text-book for the use of the United States 
Military Academy at West Point. It proved acceptable to a much wider field, and 
underwent changes and improvements, until the author's final revision, and has remained 
the standard work on the subject for many years. 

In the present edition, 1883, while the admirable features which have hitherto com- 
mended the work so highly to institutions of learning and to practical surveyors have 
been retained, some of the topics have been abridged in treatment, and some enlarged. 
Others have been added, and the whole has been arranged in the order of progressive 
development. A change which must prove particularly acceptable is the transformation 
of the article on mining-surveying into a complete treatise, in which the location of 
claims on the surface, the latest and best methods of underground traversing, &c, the 
calculation of ore-reserves, and all that pertains to the work of the mining-surveyor, 
are fully explained and illustrated by practical examples. Immediately on the publica- 
tion of this edition it was loudly welcomed in all quarters. A letter received as we 
write, from Prof. R. C. Carpenter, of the Michigan State Agricultural College, says : 
" I am delighted with it. I do not know of a more complete work on the subject, and 
I am pleased to state that it is filled with examples of the best methods ot modern 
practice. We shall introduce it as a text-book in the college course." This is a lair 
specimen of the general reception. 



Mathematical Almanac and Annual 
says : — 

" Davies is a deservedly popular author, 
%nd his mathematical works are text- 
books in many of the leading schools and 
colleges." 



Van Nostrand's Eclectic Enginuring Maga- 
zine says : — 
"We find in this new work all that can 
be asked for in a text-book. If there is a 
better work than this on Surveying, either 
for students or surveyors, our attention 
has not been called to it." 



THE NEW LEGENDRE. 

Van Amringe's Davies' Legendre. 

Elements of Geometry and Trigonometry. By Charles Davies, LL.D. Revised (18S5) 
by Prof. J. H. Van Amring*. of Columbia College. New pages. 8vo. Full leather. 

The present edition of the Legendre is the result of a careful re-examination of the 
work, into which have been incorporated such emendations in the way of greater clear- 
ness of expression or of proof as could be made without altering it in form or substance. 
Practical exercises are placed at the end of the several books, and comprise additional 
theorems, problems, and numerical exercises upon the principles of the Book or Books 
preceding. They will be found of great service in accustoming students, early in and 
throughout their course, to make for themselves practical application of geometric 
principles, and constitute, in addition, a large and excellent body of review and test 
questions for the convenience of teachers. The Trigonometry and mensuration have 
been carefully revised throughout ; the deduction of principles and rules has been sim- 
plified ; the discussion of the several cases which arise in the solution of triangles, 
plane and spherical, has been made more full and clear ; and the whole has, in definition $ 
demonstration, illustration, &c, been made to conform to the latest and best methods. .' 

It is believed that in clearness and precision of definition, in general simplicity auffl 
rigor of demonstration, in the judicious arrangement of practical exercises, in orderly 
and logical development of the subject, and in compactness of form, Davies' Legendre 
is superior to any work of its grade for the general training of the logical powers of 
pupils, and for their instruction in the great body of elementary geometric truth. 

The work has been printed from entirely new platesj and no care has been spared to 
raak© it a model of typographical excellence. 



THE .VATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 

DAYIES'S NATIONAL COURSE 
OF MATHEMATICS. 

ITS RECORD. 

In claiming for this series the first place among American text-books, of whatever 
class, the publishers appeal to the magnificent record which its volumes have earned 
during the thirty-fire yean of Dr. Charles Davies's mathematical labors. The unremit- 
ting exertions of a bfe-time have placed the modem series on the same proud eminence 
among competitors that each of its predecessors had successively enjoyed in a course of 
constantly improved editions, now rounded to their perfect fruition, — for it seems 
almost that this science is susceptible of no further demonstration. 

During the period alluded to, many authors and editors in this department have 
started into public notice, and, by borrowing ideas and processes original with Dr. Davie?, 
have enjoyed a brief popularity, but are now almost unknown. Many of the series of 
to-day, built upon a similar basis, and described as " modern books," are destined to a 
similar fate ; while the most far-seeing eye will find it difficult to fix the time, on the 
basis of any data afforded by their past history, when these books will cease to increase 
and prosper, and fix a still firmer hold on the affection of every educated American. 

One cause of this unparalleled popularity is found in the fact that the enterprise of the 
author did not cease with the original completion of his books. Always a practical 
teacher, he has incorporated in his text-books from time to time the advantages or every 
improvement in methods of teaching, and every advance in science. During all the 
years in which he has been laboring he constantly submitted his own theories and those 
of others to the practical test of the class-room, approving, rejecting, or modifying 
them as the experience thus obtained might suggest. In this way he has been able 
to produce an almost perfect series of class-books-, in which every department of 
mathematics has received minute and exhaustive attention. 

Upon the death of Dr. Davies, which took place in 1S76, his work was immediately 
taken up by his former pupil and mathematical associate of many years, Prof. W. G. 
Peck, L.L.D., of Columbia College. By him, with Prof. J. H. Van Amringe, of Columbia 
College, the original series is kept carefully revised and up to the times. 



Davies's System is the acknowledged National Standard for the United 
States, for the following reasons : — 

1st. It is the basis of instruction in the great national schools at West Point and 
Annapolis. 

2d. It has received the quasi indorsement of the National Congress. 

3d. It is exclusively used in the public schools of the National Capital. 

4th. The officials of the Government use it as authority in all cases involving mathe- 
matical questions. 

5th. Our great soldiers and sailors commanding the national armies and navies wore 
educated in this system. So have been a majority of eminent scientists in this country. 
All these refer to "Davies " as authority. 

6th. A larger number of American citizens have received their education from this 
than from any other series. 

7th. The series has a larger circulation throughout the whole country than any othw. 
being extensively used in every State in the Union. 

22 



THE NATIONAL SERIES OF STANDARD SCHOx/L-BOOKS. 
DAVIES AND PECK'S ARITHMETICS. 

OPTIONAL OR CONSECUTIVE. 

The best thoughts of these two illustrious mathematicians are combined in the 
following beautiful works, which are the natural successors of Davies's Arithmetics, 
sumptuously printed, and bound in crimson, green, and gold: — 

Davies and Peck's Brief Arithmetic. 

Also called the " Elementary Arithmetic. " It is the shortest presentation of the sub- 
ject, and is adequate for all grades in common schools, being a thorough introduction to 
practical life, except for the specialist. 

At first the authors play with the little learner for a few lessons, by object-teaching 
and kindred allurements ; but he soon begins to realize that study is earnest, as he 
becomes familiar with the simpler operations, and is delighted to hud himself master of 
important results. 

The second part reviews the Fundamental Operations on a scale proportioned to 
the enlarged intelligence of the learner. It establishes the General Principles and 
Properties of Numbers, and then proceeds to Fractions. Currency and the Metric 
System are fully treated in connection with Decimals. Compound Numbers and Ke- 
duction follow, and finally Percentage with all its varied applications. 

An Index of words and principles concludes the book, for which every scholar and 
most teachers will be grateful. How much time has been spent in searching for a half- 
forgotten definition or principle in a former lesson ! 

Davies and Peck's Complete Arithmetic. 

This work certainly deserves its name in the best sense. Though complete, it is not, 
like most others which bear the same title, cumbersome. These authors excel in clear, 
lucid demonstrations, teaching the science pure and simple, yet not ignoring convenient 
methods and practical applications. 

For turning out a thorough business man no other work is so well adapted. He will 
have a clear comprehension of the science as a whole, and a working acquaintance 
with details which must serve him well in al 1 emergencies. Distinguishing features of 
the book are the logical progression of the subjects and the great variety of practical 
problems, not puzzles, which are beneath the dignity of educational science. A clear- 
minded critic has said o." Dr. Peck's work that it is free from that juggling with 
numbers which some authors falsely call " Analysis." A series of Tables for converting 
ordinary weights and measures into the Metric System appear in the later editions. 



PECK'S ARITHMETICS. 
Peck's First Lessons in Numbers. 

This book begins with pictorial illustrations, and unfolds gradually the science of 
numbers. It noticeably simplifies the subject by developing the principles of addition 
and subtraction simultaneously ; as it does, also, those of multiplication and division. 

Peck's Manual of Arithmetic. 

This book is designed especially lor those who seek sufficient instruction to carry 
them successfully through practical life, but have not time for extended study. 

Peck's Complete Arithmetic. 

This completes the series but is a much briefer book than most of the complete 
arithmetics, and is recommended not only for what it contains, but also for what is 
omitted. 

It may be said of Dr. Peck's books more truly than of any other series published, that 
they are clear and simple in definition and rule, and that superfluous matter of every 
kind has been faithfully eliminated, thus magnifying the working value of the book 
sad saving unnecessary expense of time and labor. 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



BARNES'S NEW MATHEMATICS. 

In this series Joseph Ficklin, Ph. D., Professor of Mathematics and Astronomy 
in the University of Missouri, has combined all the best and latest results of practical 
and experimental teaching of arithmetic with the assistance of many distinguished 
mathematical authors. 



.Barnes's Elementary Arithmetic. . 
Barnes's National Arithmetic. 

These two works constitute a complete arithmetical course in two hooks. 

They meet the demand for text-books that will help students to acquire the. greatest 
amount of useful and practical knowledge of Arithmetic by the smallest expenditure of 
time, labor, and money. Nearly every topic in Written Arithmetic is introduced, and its 
principles illustrated, by exercises in Oral Arithmetic. The free use of Equations ; the 
concise method of combining and treating Properties of Numbers; the treatment of 
Multiplication and Division of Fractions in two cases, and then reduced to one ,• Can- 
cellation by the use of the vertical line, especially in Fractions, Interest, and Proportion ; 
the brief, simple, and greatly superior method of working Partia' Payments by the 
"Time Table" and Cancellation ; the substitution of formulas to l. great extent for 
rules; the full and practical treatment of the Metric System, &c, indicate their com- 
pleteness. A variety of methods and processes for the same topic, which deprive the 
pupil of the great benefit of doing a part of the thinking and labor for himself, have 
been discarded. The statement of principles, definitions \ rules, &c, is brief and simple. 
The illustrations and methods are explicit, direct, and practical. The great number 
and variety of Examples embody the actual business of the day. The very large 
amount of matter condensed in so small a compass has been accomplished by econo- 
mizing every line of space, by rejecting superfluous matter and obsolete terms, and by 
avoiding the repetition of analyses, explanations, and operations in the advanced topics 
which have been used in the more elementary parts of these books. 

AUXILIARIES. 

For use in district schools, and for supplying a text-book in advanced work for 
classes having finished the course as given in the ordinary Practical Arithmetics, the 
National Arithmetic has been divided and bound separately, as follows : — 

Barnes's Practical Arithmetic. 

Barnes's Advanced Arithmetic. 

In many schools there are classes that for various reasons never reach beyond 
Percentage. It is just such cases where Barnes's Practical Arithmetic will answer a 
good purpose, at a price to the pupil much less than to buv the complete book. On the 
other hand, classes having finished the ordinary Practical Arithmetic can proceed 
with the higher course by using Barnes's Advanced Arithmetic. 

For primary schools requiring simply a table book, and the earliest rudiments 
forcibly presented through object-teaching and copious illustrations, we have 
prepared 

Barnes's First Lessons in Arithmetic, 

which begins with the most, elementary notions of numbers, and proceeds, by simple 
steps, to develop all the fundamental principles of Arithmetic. 



Barnes's Elements of Algebra. 

This work, as its title indicates, is elementary in its character and suitable for use, 
[1J m such public schools as give instruction in the Elements of Algebra : (2) in institu- 
tions of learning whose courses of study do not include Higher Algebra ; (3) in schools 
whose object is to prepare students for entrance into our colleges and universities. 
ihis book will also meet the wants of students of Physics who require some knowledge of 



0) 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



Algebra. The student's progress in Algebra depends very largely upon the proper treat- 
ment of the four Fundamental Operations. The terms Addition, Subtraction, Multiplication, 
and Division in Algebra have a wider meaning than in Arithmetic, and these operations 
have been so defined as to include their arithmetical meaning ; so that the beginner 
is sinrply called upon to enlarge his views of those fundamental operations. Much 
attention has been given to the explanation of the negative sign, in order to remove the 
well-known difficulties in the use and interpretation of that sign. Special attention is 
here called to " A Short Method of Removing Symbols of Aggregation," Art. 76. On 
account of their importance, the subjects of Factoring, Greatest Common Divisor, and 
Least Common Multiple have been treated at greater length than is usual in elementary 
works. In the treatment of Fractions, a method is used which is quite simple, and, 
^t the same time, more general than that usually employed. In connection with Badicul 
Quantities the roots are expressed by fractional exponents, for the principles and rules 
applicable to integral exponents may then be used without modification. The Equation 
is made the chief subject of thought in this work. It is defined near the beginning, 
and used extensively in every chapter. In addition to this, four chapters are devoted 
exclusively to the subject of Equations. All Proportions are equations, and in their 
treatment as such all the difficulty commonly connected with the subject of Proportion 
disappears. The chapter on Logarithms will doubtless be acceptable to many teachers 
who do not require the student to master Higher Algebra before entering upon the 
study of Trigonometry. 



HIGHER MATHEMATICS. 
Peck's Manual of Algebra. 

Bringing the methods of Bourdon within the range of the Academic Course. 

Peck's Manual of Geometry. 

By a method purely practical, and unembarrassed by the details which rather confuse 
than simplify science. 

Peck's Practical Calculus. 
Peck's Analytical Geometry. 
Peck's Elementary Mechanics. 
Peck's Mechanics, with Calculus. 

The briefest treatises on these subjects now published. Adopted by the great Univer- 
sities : Yale, Harvard, Columbia, Princeton, Cornell, &c. 

Macnie's Algebraical Equations. 

Serving as a complsment to the more advanced treatises on Algebra, giving special 
attention to the analysis and solution of equations with numerical coefficients. 

Church's Elements of Calculus. 

Church's Analytical Geometry. 

Church's Descriptive Geometry. With plates. 2: yois. 

These volumes constitute the "West Point Course "in their several departments. 
Prof. Church was long the eminent professor of mathematics at West Point Military 
Academy, and his works are standard in all the leading colleges. 

Courtenay's Elements of Calculus. 

A standard work of the very highest grade, presenting the most elaborate attainable 
survey of the subject. 

Hackley's Trigonometry. 

With applications to Navigation and Surveying, Nautical and Practical Geometry. 
and Geodesy. 

25 



THt NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



BARNES'S ONE-TERM HISTORY 
SERIES. 



Jll^llff 




*& 



A Brief History of the United 
States. 

This is probably the most original school-book pub- 
lished for many years, in any department. A few of it? 
claims are the following : — 

i. Brevity. — The text is complete for grammar s-hool 
or intermediate classes, in 290 12mo pages, large type. 
It may readily be completed, if desired, in one term of 
study. 

2. Comprehensiveness. — Though so brief, this book 
contains the pith of all the wearying contents of the larger 
manuals, and a great deal more than the memory usually 
retains from the latter. 

3. Interest has been a prime consideration. Small 
Looks have heretofore been bare, full of dry statistics, unattractive. This one is 
charmingly written, replete with anecdote, and brilliant with illustration. 

4. Proportion of Events. — It is remarkable for the discrimination with which 
the different portions of our history are presented according to their importance. Thus 
the older works, being already large books when the Civil War took place, give it less 
space than that accorded to the Revolution. 

5. Arrangement. — In six epochs, entitled respectively, Discovery and Settlement, 
the Colonies, the Revolution, Growth of States, the Civil War. and Current Events. 

6. Catch Words — Each paragraph is preceded by its leading thought in promi- 
nent type, standing in the student's mind for the whole paragraph. 

7. Key Notes. — Analogous with this is the idea of grouping battles, &c, about 
some central event, which relieves the sameness so common in such descriptions, and 
renders each distinct by some striking peculiarity of its own. 

8. Foot-Notes. — These are crowded with interesting matter that is not strictly a 
part of history proper. They may be learned or not, at pleasure. They are certain 
in any event to be read. 

9. Biegt aphies of all the leading characters are given in full in foot-notes. 

i». Maps. — Eletrant and distinct maps from engravings on copper-plate, and beauti- 
fully «olored ; precede each epoch, and contain all the places named. 

11. Questions are at the back of the book, to compel a more independent use of the 
text. Both text and questions are so worded that the pupil must give intelligent 
answers in eis own words. " Yes " and " No " will not do. 

27 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS, 



HISTORY — Continued. 

12. Historical Recreations. — These are additional questions to test the student's 
knowledge, in review, as: "What trees are celebrated in our history?" "When 
did a fog save our army?" "What Presidents died in office?" "When was the 
Mississippi our western boundary?" "Who said, 'I would rather be right than 
President ' ? " &c. 

13. The Illustrations, about seventy in number, are the work of our best artists 
and engravers, produced at great expense. They are Advid and interesting, and mostly 
upon subjects never before illustrated in a school-book. 

14. Dates- — Only the leading dates are given in the text, and these are so associated 
as to assist the memory, but at the head of each page is the date of the event fmt 
mentioned, and at the close of each epoch a summary of events and dates. 

15. The Philosophy of History is studiously exhibited, the causes and effects 
of events being distinctly traced and their inter-connection shown. 

16. Impartiality. — All sectional, partisan, or denominational views are avoided. 
Facts are stated after a careful comparison of all authorities without the least prejudice 
or favor. 

17. Index. — A verbal index at the close of the book perfects it as a work of reference. 
It will be observed that the above are all particulars in which School Histories have 

been signally defective, or altogether wanting. Many other claims to favor it shares in 
common with its predecessors. 



TESTIMONIALS. 



From Prof. Wm. F. Allen, State Uni- 
versity of Wisconsin. 
"Two features that I like very much 
are the anecdotes at the foot of the pa,cre 
and the 'Historical Recreations' in the 
Appendix. The latter, I think, is quite 
a new feature, and the other is very well 
executed." 

From Hon. Newton Bateman, Superin- 
tendent Public Instruction, Illinois. 
" Barnes's One-Term History of the 
United States is an exceedingly attrac- 
tive and spirited little book. Its claim 
to several new and valuable features seems 
well founded. Under the form of six well- 
defined epochs, the history of the United 
States is traced tersely, yet pithily, from 
the earliest times to the present day. A 
good map precedes each epoch, whereby 
the history and geography of the period 
may be studied together, as they always 
should be. The syllabus of each paragraph 
is made to stand in such bold relief, by 
the use of large, heavy type, as to be of 
much mnemonic value to the student. The 
book is written in a sprightly and pi- 
quant style, the interest never flagging 
from beginning to end, — a rare and diffi- 
cult achievement in works of this kind." 

From HON. Abner J. Phipps, Superin- 
tendent Schools, Lewis ton, Maine. 
s * Barnes's History of the United States 



has been used for several years in the 
Lewiston schools, and has proved a very 
satisfactory work. I have examined the 
new edition of it." 

From Hon. R. K. Buchell, City Superin- 
tendent Schools, Lancaster, Pa. 

" It is the best history of the kind I have 
ever seen." 

From T. J. Charlton, Superintendent 
Public Schools, Vincennes, Ind. 
"We have used it here for six years, 
and it has given almost perfect satisfac- 
tion. . . . The notes in fine print at the 
bottom of the pages are of especial value." 

From Prof. Wm. A. Mowrv, E. $r C. 
School, Providence, R. I. 

" Permit me to express my high appre- 
ciation of your book. I wish all text- 
books for the young had equal merit." 

From Hon. A. M. Keiley, City Attorney, 
Late Mayor, and President of the School 
Board, City of Richmond, Va. 
" I do not hesitate to volunteer to you 
the opinion that Barnes 's History is en- 
titled to the preference in almost every 
respect that distinguishes a good school- 
book. . . . The narrative generally exhibits 
the temper of the judge ; rarely, if ever, 
of the advocate." 



.28 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



Primary History ot the United States. 

For Intermediate Classes. 12mo 225 
introduction to Barnes's Historical iSeries. 
From Prof. C. W. Richards, High 

School, Oswego, N.Y. 
"I think it an admirable book." 



From D. Beach, of Gibbons cO B^ach, 20 
West 59th Street, N.Y. City. 
• c The little History is to me a very 
fctractive book." 

From Prof. C. D. Larkins, Fayette- 

villc, N.Y. 
"It is the only Primary History that I 
ever saw that I liked." 

From Prof. L, R. Hopkins, U'eedsriort, 
N.Y. 
" I think Barnes's Primary History by 
far the best I ever saw." 

From Prof. Richard H. Lewis, Kingston 
College, N.C. 
" The subject matter is very good, and 
shows remarkable condensing power in 
the author." 

From Prof. Edward Smith, Supt. of 
Schools, Syracuse, N. Y. 

"It is a very interesting and pretty 
book. I should like it very much for 
supplementary reading. " 

From General Horatio C. King, 
Brooklyn, N.Y. 

"I am especially pleased Avith the new 
Primary History, which is remarkably 
concise' and interesting and free from 
partisan bias." 

From Prof S. G. Harris, Dryden, N.Y. 

" Having a few days' vacation I found 
time to carefully examine the Primary 
History you sent me and am highly de- 
lighted with it. It will satisfy a long- 
felt want." 

From the New England Jour- 
nal of Education. 

"The book is printed in 
the best type, on the finest 
paper, and is illustrated in 
the most superb, even sump- 
tuous manner. Any child 
who studies this exceptional- 
ly beautiful little book wil 
unavoidably have a higher 
regard for his country on 
account of the superior and 
charming character of the 
book." 




pages. Beautifully illustrated A fitting 

From Mr. H. H. Smith, Pi est. Beard of 

Education, Vineyard Haven, Mass. 
" I should think yon would feel proud 
of the work. " 

From Dr. Eugene Bouton, Albany, N.Y. 

" I must congratulate every one on the 
publication of this beautiful History.'' 

From Prof. II. C. Talmadge, Wood- 
bury, Ct. 

"It is the book that I have been look- 
ing for quite a long time." 

From Prof. L. C. Foster. Supt. of 

Schools, Ithaca, N. Y. 
" It is indeed a very beautiful book, 
and it seems to me well adapted for use 
in the lower grammar grades." 

From Prof. F. H. Hall, Sinclairvillc, 
N.Y. 

" This History is the best thing of the 
kind I have ever seen How it could be 
improved I do not see." 

From Prof. J. C Crfikshank, Supt. of 
Education, Fassaic Co., N.J. 
" It is the book needed, and will fill the 
gap of early historical instruction in the 
schools." 

From Prof. S. R. Morse, Supt. of Educa- 
tion, Atlantic Co. , N.J. 
"I have examined Barnes's Primary 
History of the United States and find it 
just what we have wanted in our schools. " 

FromH. E. Perkins, School Commissioner, 
Livingston Co., X.Y. 
"I 'hink it the be't Primary United 
States History that I ever examined, and 
will recommend it to my teachers." 

From The Indiana School 

Journal. 
" This book, comprised in 
223 pages, is what its title 
indicates, primary in matter 
and manner of treatment, and 
not simply an abbreviation 
of a large book. By not at- 
tempting everything there is 
space for a fuller discussion 
of the more important points. 
The author has clearly dis- 
criminated between simplicity 
of style and sinrple thought." 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 







A Brief History of An- 
cient Peoples. 



With an account of their monuments, 
literature, and manners. 340 pages. 
12mo. Profusely illustrated. 

In this work the political history, 
which occupies nearly, if not all, 
the ordinary school text, is condensed 
to the salient and essential facts, in 
order to give room for a clear outline 
the literature, religion, architecture, 
character, habits, &c, of each nation. 
Surely it is as important to know some- 
thing about Plato as all about Cajsar, 
and to learn how the ancients wrote 
their books as how they fought their 
battles. 

The chapters on Manners and Cus- 
toms and the Scenes in Real Life repre- 
sent the people of history as men and 
women subject to the same wants, hopes 
and fears as ourselves, and so briag the distant past near to us. The Scenes, which are 
intended only for reading, are the result of a careful study of the unequalled collections of 
monuments in the London and Berlin Museums, of the ruins in Rome and Pompeii, and 
of the latest authorities on the domestic life of ancient peoples. Though intentionally 
written in a semi-romantic style, they are accurate pictures of what might have occurred, 
and some of them are simple transcriptions of the details sculptured in Assyrian 
alabaster or painted on Egyptian walls. 

29 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



HISTORY— Continued. 

The extracts made from the sacred books of the East are not specimens of their style 
and teachings, but only gems selected often from a mass of matter, much of which would 
be absurd, meaningless, and even revolting. It has not seemed best to cumber a book 
like this with selections conveying no moral lesson. 

lhe numerous cross-references, the abundant dates in parenthesis, the pronunciation 
of the names in the Index, the choice reading references at the close of each general 
subject, and the novel Historical Recreations in the Appendix, will be of service to 
teacher and pupil alike. 

Though designed primarily for a text-book, a large class of persons —general readers, 
who desire to know something about the progress of historic criticism and «ne recent 
discoveries made among the resurrected monuments of the East, but have no leisure to 
read the ponderous volumes of Brugsch, Layard, Grote, Mommsen, and lime — will hud 
this volume just what they need. 



From Homer B. Spbague, Heal Master 
Girls' High School, West Newton St., Bos- 
ton, Mass. 
" I beg to recommend in strong terms 

the adoption of Barnes's 'History of 



Ancient Peoples ' as a text-book. It is 
about as nearly perfect as could be 
hoped for. The adoption would give 
great relish to the study of Ancient 
History." 



l&B&Sjl 




HE Brief History of France. 



2§§ 



By the author of the " Bri-.f United States," 
with all the attractive features of that popu- 
lar work (which see) and new ones of its own. 
It is believed that the History of France 
has never before been presented in such 
brief compass, and this is effected without 
i\ * Y;^^V~Mt> ^"^t^fip^ sacrificing one particle of interest. The book 

reads like a romance, and, while drawing the 
student by an irresistible fascination to his 
task, impresses the great outlines indelibly upon the memory. 

30 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 



HISTORY — Continued. 

Barnes's Brief History of Mediaeval and Modern 
Peoples. 

The success of the History of Ancient Peoples was immediate and great. A History 
of Mediaeval and Modern History, upon the same plan, was the natural sequence. 
Those teachers who used the former will be glad to know that the latter book is now 
ready, and classes can go right on without changing authors. 

The New York School Journal says : — 

" The fine-print notes . . . work a field 
not widely developed until Green's His- 



tory of English People appeared, relating 
to the description of real, every-day life 
of the people." 

This work distinguishes between the period of the world's history from the Fall of 
Rome (a.d. 476) to the Capture of Constantinople (a.d. 1453), — about one thousand 
years, called "Middle Ages," — and the period from the end of the fifteenth century to 
the present time. It covers the entire time chronologically and by the order of events, 
giving one hundred and twenty-two fine illustrations and sixteen elaborate maps. 




[Illustration from Barnes's Brief - History 
Series.] 

The subject has never before been so in- 
terestingly treated in brief compass. The Po- 
litical History of each nation is first given, 
theii the Manners and Customs of the People. 
A better idea of the growth of civilization and 
the chances in the condition of mankind can- 
not be found elsewhere. The book is fitted 
for private reading, as well as schools. 

31 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS, 

HISTORY — Continued. 

Barnes's Brief General History. 

Comprising Ancient, Mediaeval, and Modern Peoples. 

The special features of this book are as follows: — 

The General History contains 600 pages. Of this amount, 350 pages are devoted to 
the political history, and 250 pages to the civilization, manners, and customs, etc. The 
latter are in separate chapters, and if the time of the teacher is limited, may be omitted. 
The class can thus take only the political portion when desired. The teacher will have, 
however, the satisfaction of knowing that, such is the fascinating treatment of the 
civilization, literature, etc. , those chapters will be carefully read by the pupils ; and, on 
the principle that knowledge acquired from love alone is the most vivid, will probably 
be the best-remembered part of the book. This portion of the book is therefore all 
clear gain. 

The Black-board Analysis. See p. 314 as an example of this marked feature. 

The exquisite Illustrations, unrivalled by any text-book. (See pp. 9,457, and 5S2, as 
samples of the 240 ruts contained in this beautiful work. 

The peculiar Summaries, and valuable lists of Reading References. See p. 417. 

The numerous and excellent colored Maps. These are so full as to answer for an 
extensive course of collateral reading, and are consequently useful for reference outside 
of class-work. See pp. 299 and 317- 

The Scenes in Real Life, which are the result of a careful study of the collections 
and monuments in the London, Paris, and Berlin museums, and the latest authorities 
upon the domestic life of the people of former times. See pp. 3S-39. This scene — 
a Lord of the IYth Dynasty — is mainly a transcription of details to be found painted 
on the walls of Egyptian tombs. 

The chapters on Civilization that attempt to give some idea of the Monuments, Arts, 
Literature, Education, and Manners and Customs of the different nations. See pp. 171, 
ISO, 276, 279, 472, and 514. 

The admirable Genealogical Tables interspersed throughout the text. See pp. 340 
and 494. 

The Foot-Notes that are packed full of anecdotes, biographies, pleasant information, 
and suggestive comments. As an illustration of these, take the description of the 
famous sieges of Haarlem and Leyden, during the Dutch War of Independence, w 446 
and 443. ' **." 

The peculiar method of treating Early Roman History, by putting in the text the 
facts as accepted by critics, and, in the notes below, the legends. See pp 205-6. 

The exceedingly useful plan of running collateral history in parallel columns, as for 
example on p. 361, taken from the Hundred Years' War. 

The Historical Recreations, so valuable in arousing the interest of a class See 
p. xi from the Appendix. 

The striking opening of Modern History on pp. 423-4 

The interesting Style, that sweeps the reader'along as by the fascination of a novel. 
The pupil insensibly acquires a taste for historical reading,' and foreets the tediousness 
of the ordinary lesson in perusing the thrilling storv of the past Sep pp 251-2 

Special attention is called to the chapter entitled Rise of Modern Nations, - 
England France, and Germany. The characteristic feature in the medieval history of 
each of these nations is made prominent, (a.) After the Four Conquests of En-land, 
the central idea in the growth of that people was the Development of Constitutional 
Liberty. (&.) lhe feature of French history was the conquest of the great vassals by 

J™§ « rv trium P h of royalty over feudalism, and the final consolidation of the 
scattered nets into one grand monarchy, (c.) The characteristic of German history was 
disunion emphasized by the lack of a centr.nl capital city, and by an elective rather than 
an hereditary monarchy. The struggle of the Crown with its powerful vassals was the 
simeas in France, but developed no national sentiment, and ended in the establishment 
of semi-independent dukedoms. 

These three thoughts furnish the beginner with as many threads on which to string 
the otherwise isolated facts of this bewildering period. 



32 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 

BOOK-KEEPING. 

Powers's Practical Book-keeping. 
Powers's Blanks to Practical Book-keeping. 

A Treatise on Book-keeping, for Public Schools and Academies. By Millard R. 
Powers, M. A. This work is designed to impart instruction upon the science of accounts, 
as applied to mercantile business, and it is believed that more knowledge, and that, too', 
of a more practical nature, can be gained by the plan introduced in this w«k, than by 
any other published. 

Folsom's Logical Book-keeping. 
Folsom's Blanks to Book-keeping. 

This treatise embraces the interesting and important discoveries of Professor Folsom (of 
the Albany " Bryant & Stratton College "), the partial enunciation of which in lectures 
and otherwise has attracted so mucii attention in circles interested in commercial 
education. 

After studying business phenomena for many years, he has arrived at the positive 
laws and principles that underlie the whole subject of accounts ; finds that the science 
is based in value as a generic term ; that value divides into two classes with varied 
species ; that all the exchanges of values are reducible to nine equations ; and that all 
the results of all these exchanges are limited tu thirteen in number. 

As accounts have been universally taught hitherto, without setting out from a radical 
analysis or definition of values, the science has been kept in great obscurity, and been 
made as difficult to impart as to acquire. On the new theory, however, these obstacles 
are chiefly removed. In reading over the first part of it, in which the governing laws 
and principles arc discussed, a person with ordinary intelligence will obtain a fair con- 
ception of the double-entry process of accounts. But when he comes to study thoroughly 
these laws and principles as there enunciated, and works out the examples and memo- 
randa which elucidate the thirteen results of business, the student will neither fail in 
readily acquiring the science as it is, nor in becoming able intelligently to apply it in 
the interpretation of business. 

Smith and Martin's Book-keeping. 
Smith and Martin's Blanks. 

This work is by a practical teacher and a practical book-keeper. It is of a thoroughly 
popular class, and will be welcomed by every one who loves to see theory and practice 
combined in an easy, concise, and methodical form. 

The single-entry portion is well adapted to supply a want felt in nearly all other 
treatises, which seem to be prepared mainly for the use of wholesale merchants ; 
leaving retailers, mechanics, farmers, &c. , wiio transact the greater portion of the 
business of the country, without a guide. The work is also commended, on this 
account, for general use in young ladies' seminaries, where a thorough grounding 
in the simpler form of accounts will be invaluable to the future housekeepers of the 
nation. 

The treatise on double-entry book-keeping combines all the advantages of the 
most recent methods with the utmost simplicity of application, thus affording the 
pupil all the advantages of actual experience in the counting-house, and giving a 
clear comprehension of the entire subject through a judicious course of mercantile 
transactions. 

PRACTICAL BOOK-KEEPING. 

Stone's Post-Office Account Book. 

By Micah H. Stone. For record of Box Rents and Postages. Three si";es always in 

stock. 64, 10S, and 204 pages. 

INTEREST TABLES. 

Brooks's Circular Interest Tables. 

To calculate simple and compound interest for any amount, from 1 cent to $1,000, at 
current rates from 1 day to 7 years. 

41 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 

DR. STEELE'S ONE-TERM SERIES* 
IN ALL THE SCIENCES. 

Steele's 14-Weeks Course in Chemistry. 
Steele's 14-Weeks Course in Astronomy. 
Steele's 14-Weeks Course in Physics. 
Steele's 14-Weeks Course in Geology. 
Steele's 14-Weeks Course in Physiology. 
Steele's 14-Weeks Course in Zoology. 
Steele's 14-Weeks Course in Botany. 

Our text-books in these studies are, as a general thing, dull and uninteresting. 
They contain from 400 to U00 pages of dry facts and unconnected details. They abound 
in that which the student cannot learn, much less remember. The pupil commences 
the study, is confused by the tine print and coarse print, and neither knowing exactly 
what to learn nor what to hasten over, is crowded through the single term generally 
assigned to each branch, and frequently comes to the close without a definite and exact 
idea of a single scientific principle. 

Steele's " Fourteen- Weeks Courses " contain only that which every well-informed per- 
son should know, while all that which concerns only the professional scientist Is omitted. 
The language is clear, simple, and interesting, and the illustrations bring the subject 
within the range of home life and daily experience. They give such of the general 
principles and the prominent facts as a pupil can make familiar as household words 
within a single term. The type is large and open; there is no fine print to annoy ; 
the cuts are copies of genuine experiments or natural phenomena, and are of fine 
execution. 

In fine, by a system of condensation peculiarly his own, the author reduces each 
branch to tlie limits of a single term of study, while sacrificing nothing that is essential, 
and nothing that is usually retained from the study of the larger manuals in common 
use. Thus the student lias rare opportunity to economize his lone, or rather to employ 
that which he has to the best advantage. 

A notable feature is the author's charming "style," fortified by an enthusiasm over 
his subject in which the student will not fail to partake. Believing that Natural 
Science' is full of fascination, he has moulded it into a form that attracts the attention 
and kindles the enthusiasm of the pupil 

The recent editions contain the author's "Practical Questions" on a plan never 
before attempted in scientific text-books. These are questions as to the nature and 
cause of common phenomena, and are not directly answered in the text, the design 
being to test and promote an intelligent use of the student's knowledge of the foregoing 
principles. 

Steele's Key to all His Works. 

This work is mainly composed of answers to the Practical Questions, and solutions of the 
problems, in the author's celebrated " Fourteen-Weeks Courses " in the several sciences, 
with many hints to teachers, minor tables, &c. Should be on every teacher's desk. 

Prof. J. Dorman Steele is an indefatigable student, as well as author, and his books 
have reached a fabulous circulation. It is safe to say of his books that they have f 
accomplished more tangible and better results in the class-room than any other ever 
offered to American schools, and have been translated into more languages for foreign 
sehools. They are even produced in raised type for the blind. 



THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. 

NATURAL SCIENCE — Continued. 

TEMPERANCE PHYSIOLOGY. 

Steele's Abridged Physiology, for Common Schools. 
Steele's Hygienic Physiology, for High Schools. 

With especial reference to alcoholic drinks and narcotics. Adapted froln " Fourteen 
Weeks' Course in Human Physiology. " By J. Dorman Steele, Ph.D. Edited and 
Bndorsed for the use of schools (in accordance with the recent legislation upon tins 
subject) by the Department of Temperance Instruction of the W. (J. T. U. of the Uuited 
States, under the direction of Mrs. Mary H. Hunt, superintendent. 

This new work contains all the excellent and popular features that have given Dr. 
Steele's Physiology so wide a circulation. Among these, are the following: 

1. Colored Lithographs to illustrate the general facts in Physiology. 

2. Black-board Analysis at the beginning of each chapter. These have been 
found of great service, in class-work, especially' in review and examination. 

3. The Practical Questions at the close of each chapter. These are now too well 
known to require any explanation. 

4. The carefully prepared sections upon the Physiological Action of Alcohol, 
Tobacco, Opium, etc. These are scattered through the book as each organ is treated. 
This subject is examined from a purely scientific stand-point, and rejjresents the latest 
teachings at home and abroad. While there is no attempt to incorporate a temperance 
lecture in a school-book, yet the terrible effects of these " Stimulants and Narcotics," 
especially upon the young, are set forth all the more impressively, since the lesson is 
taught merely by the presentation of facts that lean toward no one's prejudices, and 
admit of no answer or escape. 

5. Throughout the book, there are given, in text and foot-note, experiments that can 
be performed by teacher and pupil, and which, it is hoped, will induce some easy dis- 
sections to be made in every class, and lead to that constant reference of all subjects to 
Nature herself, which is so invaluable in scientific study. 

6. The collection of recent discoveries, interesting facts, etc., in numerous foot- 
notes. 

7. The unusual space given to the subject of Ventilation, which is now attracting 
so much attention throughout the country. 

8. The text is brought up to the level of the new Physiological views. The division 
into short, pithy paragraphs ; the bold paragraph headings ; the clear, large type ; the 
simple presentation of each subject ; the interesting style that begets in every child a 
love of the study, and the beautiful cuts, each having a full scientific description and 
nomenclature, so as to present the thing before the pupil without cumbering the text 
with the dry details, — all these indicate the work of the practical teacher, and will be 
appreciated in every school-room. 

Child's Health Primer. 

For the youngest scholars. 12mo, cloth, illustrated. 

Hygiene for Young People. 

Prepared under the supervision of Mrs. Mary H. Hunt, Superintendent of the 
Department of Scientific Instruction of the " Women's National Christian Temperance 
Union." Examined and approved by A. B. Palmer, M.D., University of Michigan. 

Jarvis's Elements of Physiology, 
jarvis's Physiology and Laws of Health. 

Tiie only books extant which approach this subject with a proper view of the true 
object of teaching Physiology in schools, viz. , that scholars may know how to take care 
of' their own health. In bold contrast with the abstract Anatomies, which children 
learn as they would Greek or Latin (and forget as soon), to discipline the mind, are these 
text-books, using the science as a secondary consideration, and only so far as is neces- 
sary for the comprehension of the laws of health. 



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